by osborne reynolds, ll,d,, fm.s. contents. · 2011. 8. 2. ·...

157 ]

IV. On the Theory of Lubrication and its Application to Mr. Beauoha.mp Toweb's

Experiments^ including an Experimental Determination of the Viscosity of Olive Oil,

By Professor Osborne Reynolds, LL,D,, FM.S.

Received December 29, 1885,—Bead February 11, 1886.

[Plate 8.]

CONTENTS.

Section I.

—

Inteoductory.ARTICLE.

Discordance of experimental results ,......., 1

Mr. Tower's discovery of the separating film of oil, &c 2

The idea of a bydrodynamical theory of lubrication ...,..,, 3

The equation of lubrication mentioned before Section A, Montreal, and subsequently

integrated 4

The comparison of the theoretical results with experimental shows a temperature effect 5

Detei^mination of the variation of viscosity of olive oil brings the theory into complete

accordance with experiments, and shows how various circumstances affect the

results 6

The difference in the radii of brass and journal, and the point of nearest approach of

brass to journal, and explanation of increased heating on first reversal... ........ 7

The limits of complete lubrication, incomplete lubrication, and necking 7a

The general arrangement of the paper 8

Section II.—The Properties op Lijbbioants.

Definition of viscositv ..,...........,.,.,., 9

The character of viscosity ..,,..,,..., 10

The two viscosities ^ ....... , 11

Experimental determination of the value of /i for olive oil, &c., figs^ 2 and 8, Table I. 12;

The comparative values for /* for different fluids and different units ,13

Section III.

—

Generai^ View of the Action op Lubrication.

The c^se of two nearly parallel surfaces separated by a viscous fluid. ,,...,,., 14

The Case of revolving cylindrical surface r ...-. f ••• . 1^

The effect of limited supply of lubricating material 16

The relation between resistance, load, and speed for limited lubrication 17

The conditions of equilibrium with cylindrical surfaces 17a

The wear and heating of bearings 18

158 PKOFESSOE O. EEYI^OLDS ON THE THEORY OF LUBRICATION

Section IV.

—

The Equations of Hydeodynamics as Applied to Lubrication.ARTICLE!.

The complete equations for interior of viscous fluid simplified , 19

The boundary conditions 20

The first integration of the resulting equations—equations of lubrication 21

The conditions under which further integration has been undertaken 22

Section V.

—

Application to Oases in which the Equations are Completely Integrable.

Parallel plane surfaces approaching each other, the surfaces having elliptical boundaries 23

Plane surfaces of unlimited length ,.,.....,..,,.,*,,, 24

Section YL—The Integration of the Equations for Cylindrical Surfaces.

General adaptation of the equations 25

The method of approximate integration , , , 26

Integration of the equations , . , . » 27

Section YII,—Solution of the Equations for Cylindrical Surfaces.

c and a/ ^ small compared with unity. , , . . . , 28

Further approximation to the solution of the equations for particular values of c,

Plate 8, figs. 1, 2, and 3 ......... , , . . 29

<?='5 the limit to this method of integrating ,...,.... 29a

Section Vlll.—-Tee Effects of Heat and Elasticity.

jii and a are only to be inferred from experiments 30

The effect of the load and the velocity to alter a , 31

The effect of speed on the temperature ......,,...,...,........,.., 32

The formulae for temperature and friction, and interpretation of constants ,,..,,.,.. 33

The maximum load at any speed ,...,, 33a

Section IX.

—

Application op the Equations to Mr. Tower's Experiments.

Eeferences to Mr, Tower's reports, Tables I., IX., and XII 34

The effect of necking the journal 35

First approximation to the difference in the radii of the journal and brass ISTo. 1 36

The rise in temperature of the film, owing to friction , , 37

The actual temperature of the film , 38

The variation of a with the load 39

Application of the equations to the circumstances of Mr. Tower's experiments,

Tahlp TV Ad

The velocity of maximum carrying power 41

Application of the equations to determine the oil pressure with brass No. 2 42

Conclusion 43

AND ITS APPLICATION TO MR. B. TOWER'S EXPERIMENTS, 159

Section I.

—

[ntroductory.

1. Lubrication, or the action of oils and other viscous fluids to diminish friction

and wear between sohd surfaces, does not appear to have hitherto formed a subject for

theoretical treatment. Such treatment may have been prevented by the obscurity of

the physical actions involved, which belong to a class as yet but little known, namely,

the boundary or surface actions of fluids ; but the absence of such treatment has also

been owing to the want of any general laws discovered by experiment.

The subject is of such fundamental importance in practical mechanics, and the

opportunities for observation are so frequent, that it may well be a matter of surprise

that any general laws should have for so long escaped detection.

Besides the general experience obtained, the friction of lubricated surfaces has been

the subject of much experimental investigation by able and careful experimenters.

But, although in many cases empirical laws have been propounded, these fail for the

most part to agree with each other and with the more general experience.

2. The most recent investigation is that of Mr. Beauchamp Tower, undertaken at

the instance of the Institution of Mechanical Engineers. Mr. Tower's first report

was published November, 1883, and his second report in 1884 (" Proc. Inst. Mechanical

Engineers ^').

In these reports Mr. Tower, making no attempt to formulate, states the results of

experiments apparently conducted with extreme care and under very various and

well-chosen circumstances. Those results which were obtained under the ordinary

conditions of lubrication so far agree with the results of previous investigators as to

show a want of any regularity. But one of the causes of this want of regularity,

irregularity in the supply of the lubricant, appears to have occurred to Mr. Tower

early in his investigation, and led him to include amongst his experiments the unusual

circumstances of surfaces completely immersed in oil. This was very fortunate, for

not only do the results so obtained show a great degree of regularity, but while

making these experiments he was accidentally led to observe a phenomenon which,

taken with the results of his experiments, amounts to a crucial proof that in these

experiments with the oil bath the surfaces were completely and continuously sepa-

rated by a film of oil ; this film being maintained by the motion of the journal,

although the pressure in the oil at the crown of the bearing was shown by actual

measurement to be as much as 625 lbs. per sq. inch above the pressure in the oil bath.

These results obtained with the oil bath are very important, notwithstanding that

the condition is not common in practice. They show that with perfect lubrication a

definite law of variation of the friction with the pressure and velocity holds for a

particular journal and brass. This strongly implies that the irregularity previously

found was due to imperfect lubrication. Mr. Tower has brought this out :—Sub-

stituting for the bath an oily pad, pressed against the free part of the journal, and

making it so slightly greasy that it was barely perceptible to the touch, he again

160 PEOFESSOR O. RBYHOLDS OF THE THEORY OF LUBRIOATIOF

found considerable regularity in the results ; these, however, were very different from

those with the bath. Then with intermediate lubrication he obtained intermediate

results, of which he says :—^'Indeed, the results, generally speaking, were so

uncertain and irregular that they may be summed up in a few words. The friction

depends on the quantity and uniform distribution of the oil, and may be anything

between the oil bath results and seizing, according to the perfection or imperfection

of the lubrication/'

3. On reading Mr. Tower's report it occurred to the author as possible that in the

ease of the oil bath the film of oil might be sufficiently thick for the unknown

boundary actions to disappear, in which case the results would be deducible from the

equations of hydrodynamics. Mr. Tower appears to have considered this, for he

remarks that according to the theory of fluid friction the resistance would be as the

square of the velocity, whereas in his results it does not increase according to this

law. Considering how very general the law of resistance as the square of the

velocity is with fluids, there is nothing remarkable in the assumption of its holding in

Buch a case. But the study of the behaviour of fluid in very small channels, and

particularly the recent determination by the author of the critical velocity at which

this law changes from that of the square of the velocity to that of the simple ratio,

shows that with such highly viscous fluids as oils, such small spaces as those existing

between the journal and its bearing, and such limited velocity as that of the surface

of the journal, the resistance would vary, ccBteris paribus, as the velo(;ity. Further^

the thickness of the oil film would not be uniform and might be affected by the

velocity, and as the resistance would vary, cceteris paribus, inversely as the thickness

pf the film, the velocity might exert in -this way a secondary effect on the resistance ;

and, further still, the resistance would depend on the viscosity of the oil, and this

depends on the temperature. But as Mr. Tower had been careful to make all his

experiments in the same series with the journal at a temperature of 90^ Fahr., it did

not at first appear that there could be any considerable temperature effect in his

results.

4. The application of the hydrodynamical equations to circumstances similar in so

far as they were known to those of Mr. Tower's experiments, at once led to an

equation between the variation of pressure over the surface and the velocity, which

equation appeared to explain the existence of the film of oil at high pressure. This

equation was mentioned in a paper read before Section A. of the British Association

at Montreal, 1884. It also appears from a paragraph in the President's Address

(Brit, Assoc. Rep., 1884, p. 14) that Professor Stokes and Lord Raylbigh had

simultaneously arrived at a similar result. At that time the author had no idea of

attempting its integration. On subsequent consideration, however, it appeared that

the equation might be transformed so as to be approximately integrated, and the

theoretical results thus definitely compared with the experimental.

5. The result of this comparison was to show that with a particular journal and

AND ITS APPLICATION TO MR. B. TOWBR^S EXPERIMENTS. 161

brass the mean thickness of the film of oil would be sensiblv constant, and hence, if

the viscosity was constant, the resistance would increase directly as the speed. As

this was not in accordance with Mr. Tower's experiments, in which the resistance

increased at a much slower rate, it appeared that either the boundary actions became

sensible or that there must have been a rise in the temperature of the oil which had

escaped the thermometers used to measure the temperature of the journal.

That there would be some excess of temperature in the oil film on which all the

work of overcoming the friction is spent is certain ; and after carefully considering

the means of escape of this heat, it seems probable that there would be a difference of

several degrees between the oil bath and the film of oil.

This increase of temperature would be attended by a diminution of viscosity, so

that as the resistance and temperature increased with the velocity the viscosity

would diminish and cause a departure from the simple ratio >

6. In order to obtain a quantitative estimate of these secondary effects, it was

necessary to know exactly the relation between the viscosity and temperature of the

lubricant used. For this purpose an experim.ental determination was made of the

viscosity of olive oil at different temperatures as compared with the known viscosity

of water. From the results of these experiments an empirical formula has been

deduced, by means of which definite expressions have been obtained for the approxi-

mate variation of the viscosity with the speed and load. Taking these variations of

viscosity into account, the results obtained from the hydrodynamical theory are

brought into complete accordance with these experiments of Mr. Tower. Thus we

have not only an explanation of the very novel phenomena brought to light by these

experiments, and what appears to be an important verification of the assumptions on

which the theory of hydrodynamics is founded, but we also find, what is not shown in

the experiments, how the various circumstances under which the experiments have

been made affect the results.

7. Two circumstances particularly are brought out in the theory as principal

circumstances which seem to have hitherto entirely escaped notice, even that of

Mr. Tower.

One of these is the difference in the radii of the journal and of the brass or bearing.

It is well known that the fitting between the journal and its bearing produces a

great effect on the carrying power of the journal, but this fitting is rather supposed to

be a matter of smoothness of surface than a degree of correspondence in radii. The

radius of the bearing must always be as much larger than that of the journal as is

necessary to secure an easy fit ; but more than this, I think, has never been suggested.

Now it appears from the theory that if viscosity were constant the friction would

be inversely proportional to the difference in radii of the journal and the bearing, and

this although the arc of contact is less than the semicircumference. Taking the

temperature into account, it appears from the comparison of the theoretical results

with the experimental that at a temperature of 70*5^ Fahr. the radius of one of the

MDCCCLXXXVI. Y

162 PROFESSOR 0. REYNOLDS OIST THE THEORY OF LUBR1CATI0:N^

brasses used was '00077 inch greater than that of the journal, while at a temperature

of 70*^ Fahr. that of the other was '00084 inch, or 9 per cent, larger than the first.

These two brasses were probably both bedded to the journal in the same way, and

had neither of them been subjected to any great amount of wear, so that there is

nothing surprising in their being so nearly the same fit. It would be extremely

interesting to find whether prolonged wear of the brass tends to preserve or destroy

the fit- This does not appear from Mr. Tower's experiments. It does appear,

however, that the brass expands wnth an increase of temperature more than the

journal, and that its radius increases as the load increases in a very definite manner.

Another circumstance brought out by the theory, and remarked on both by Lord

Rayleigh and the author at Montreal, but not before expected, is that the point of

nearest approach of the journal to the brass is not by any means in the line of the

load, and, what is still more contrary to common supposition, is on the o^^ side of

the line of load.

This circumstance, the reason for which is rendered perfectly clear by the conditions

of equilibrium, at once accounts for a singular phenomena mentioned by Mr. Tower,

viz., that the journal having been run in one direction until the initial tendency to

heat had entirely disappeared, on being reversed it immediately began to heat again

;

but this effect stopped when the process had been often repeated. The fact being

that running in one direction the brass had been worn to the journal only on the off

side for that direction, so that when the motion was reversed the new o^ side was

like a new brass.

7a. The circumstances which determine the greatest load which a bearing will

carry with complete lubrication, ^.6., with the film of oil extending between brass and

journal throughout the entire arc, are definitely shown in the theory.

The effect of increasing the load beyond a certain small value being to cause the

brass to approach nearer to the journal at a point H which moves from A towards as

the load increases, and when the load is such that the least separating distance ca is

about half the difference of radii a, the angular position of H is 40° to the off side of O,

the middle of the brass. At this point the pressure in the oil film is everyw^here greater

than at A and B, the extremities of the brass, but when the load further increases the

pressure towards A on the off side becomes smaller or negative. This when sufficient

'^ " On" and " off " sides of the line of load are used by Mr. Tower to express respectively the sides of

approach and succession, as B and A in the figure, the arrow indicating the direction of rotation.

AKD ITS APPLIOATIOF TO MR. B. TOWER'S EXPERIMENTS. 163

will cause rupture in the oil film, which will then only bextend etween the brass and

journal over a portion of the whole arc and a smaller portion as the load increases.

Thus, since the amount of negative pressure w^hich the oil will bear depends on

circumstances which are uncertain, the limit of the safe load for complete lubrication

is that which causes the least separating distance to be half the difference of radii of

the brass and journal.

The rupture of the oil film does not take place at the point of nearest approach,

and hence the brass may still be entirely separate from the journal, and could the

integrations be effected it would be possible to deal as definitely with this condition

as with that of complete lubrication; but these difficulties have limited the actual

application of the theory to complete lubrication. This however by no means requires

an oil bath, but merely sufficient oil on the journal.

What happens when the supply of oil is limited, ^.6., insufficient for complete

lubrication, cannot be definitely expressed without further integrations ; but sufficient

may be seen to show that the brass will still be completely separated from the journal,

although the separating film will not touch the brass, except over a limited area ; but

in this case it is easy to show by general reasoning that in the one extreme, where the

supply of oil is limited, the friction increases directly as the load and is independent of

the velocity, while in the other, where the oil is abundant, the circumstances are those

of the oil bath.

The effect of the limited length of the journal is also apparent in the equations, as

is also the effect of necking the shaft to form the journal, so that the ends of the brass

are asfainst flanges on the shaft.

The theory is perfectly applicable to cases in which the direction of the load on

the bearing, varies, as wdth the crank pin and with the bearings of the crank shaft of

the steam-engine ; but these cases have not been considered, as there are no definite

experiments to compare.

8. Although in the main the present investigation has been directed to the

circumstances of Mr. Tov^er's experiments, viz., a cylindrical journal revolving in a

cyHndrical brass, it has, on the one hand, been found necessary to proceed from the

general equations of equilibrium of viscous fluids, and, on the other hand, to consider

somewhat generally the physical property of viscosity and its dependence on

temperature.

The property of viscosity has been discussed at length in Section II. ; which section

also contains the account of an experimental investigation as to the viscosity of olive

oil.

The general theory deduced from the hydrodynamical equations for viscous fluids,

with the methods of application, is given in Sections IV., V., YL, VII., and VIII.

As there are some considerations which cannot be taken into account in the more

general method, which method also tends to render obscure the more immediate

purpose of the investigation, a preliminary discussion of the problem^ illustrated by

Y 2

164 PEOFESSOE 0. EEYKOLDS OK THE THEOEY OF LUBEICATION"

aid cf the graphic method, has been introduced as Section III. Finally, the definite

application of the theory to Mr. Tower's experiments is given in Section IX.

Sectfon II.

—

The Peoperties of Lubricants.

9. The Definition of Viscosity.

In distinguishing between solid, and fluid matter, it is customary to define fluid as

a state of matter incapable of sustaining tangential or shearing stress. This defini-

tion, however, as is well known, is only true as applied to actual fluids when at rest.

The resistance encountered by water and all known fluids flowing steadily along

parallel channels, affords definite proof that in certain states of motion all actual fluids

will sustain shearing stress. These actual fluids are, therefore, called in the language

of mathematics imperfect or viscous fluids.

In order to obtain the equations of motion of such fluids, it has been necessary to

define clearly the property of viscosity. This definition has been obtained from the

consideration that to cause shearing stress in a body it is necessary to submit it to

forces tending to change its shape. Forces tending to cause a general motion, whether

linear, revolving, uniform expansion, or uniform contraction, call forth no shearing

stress.

Using the term distortion to express change of shape, apart from change of position,

uniform expansion, or contraction, the viscosity of a fluid is defined as the shearing

stress caused in the fluid while undergoing distortion, and the shearing stress divided by

the rate of distortion is called the coefficient of viscosity, or, commonly, the viscosity

of the fluid.

This is best expressed by considering a mass of fluid bounded by two parallel pla,nes

at a distance a, and supposing the fluid between these planes to be in motion in a

direction parallel to these surfaces with a velocity which varies uniformly from at one

of these surfaces to u at the other. Then the rate of distortion is

?^

a

and the shearing stress on a plane parallel to the motion is expressed by

f==i^7 • 0)'a

p. being the coefficient of viscosity or the modulus of the resistance to distortional

motion.

10. The Character of Viscosity,

In dealing wdth ideal fluids/ it is of course allowable to consider />t as being zero or

having any conceivable value ; but practically, as regards natural philosophy, the

AND ITS APPLICATION TO MR. B. [TOWER'S EXPERIMENTS. 165

value of any such considerations depends on whether the calculated behaviour of the

ideal fluid is found to agree with the behaviour of the actual fluids—whether taking

a particular fluid, a value of /x can be found such that the values of/ calculated by-

equation (1) agree with the values of f determined by experiments for all values of

a and u.

In the mathematical theory of viscous fluids /x is assumed to be constant for a

particular fluid. This supposition is sometimes justified by reference to some assumed

dynamical constitution of fluids ; but apart from such hypotheses there is no more

ground for supposing a constant value for /^ than there is for supposing a particular law

of gravitation, in other words, there is no ground at all. If a particular value of

/x. is found to bring the calculated results into agreement with all experimental results,

then this value of /x defines a property of actual fluids, and of course it has been with

this object that the mathematical theory of \l has been studied.

The chief question as regards /x is a simple one— within a particular fluid is \l

constant ? In other words, is viscosity a property of a fluid like inertia which is

independent of its motion ? If it is, our equations may be useful ; if it is not then

the introduction of /x into the equations renders them so complex that it is almost

hopeless to expect anything from them.

Another question of scarcely less practical importance relates to the character

of /x near the bounding surfaces of the fluid. If /x is constant in the fluid, does it

change its value near the boundary of the fluid ? Is there anything like sHpping

between the fluid and a solid boundary with which it is in contact ?

As regards the answers to these questions the present position is somewhat as

follows :

—

11. The Two Viscosities.

The general experience that the resistance varies as the square of the velocity is an

absolute proof that /x is not constant unless a restricted meaning be given to the

definition of viscosity, excluding such part of the resistance as may be due, in the

way explained by Prof. STOKES,'''\to internal eddies or cross streams, however insensible

these may be, so long as they are not simply molecular motions.

On the other hand in the definite experiments made by Colomb, and particularly

by PoiSEUiLLE, it was found that the resistance w^as proportional to the velocity, and

therefore that /x was absolutely constant—^.e., independent of the velocity.t

To meet this discordance it has been supposed that /x varied with the rate of distor-

tion

—

i.e., is a function of u/a, but is sensibly constant when u/a is small. J

To assume this, however, is to neglect Poisexjille^s experiments, in which he found

for water the resistance absolutely proportional to the velocity in a tube *6 mm.

* Stokes's Reprint, vol. i., p. 99.

t Paris Mem. Savans Etrang., torn. 9 (1846), p. 4B4.

X Lamb's " Motion of Eluids," 1879, Art. 180.

166 PROFESSOR O. RBYKOLDS ON THE THEORY OF LUBRIOATIOlSr

diameter up to a velocity of 6 metres per second, which corresponds to a value of

u/a=20,000.

On the other hand it is found by Da ROY^ and others in large tubes that the

resistance varies as the square of the velocity for values of-, as low as L Thus in a

tube of '6 mm. we have ft constant for all rates of distortion below 20,000, while in a

tube of 500 mm. diameter ju. is a function of the distortion for all values greater

thanl.

It is, therefore, clear that if fju is a function of the distortion it must also be a func-

tion of the dimensions of the channels, and in that case jjl cannot be considered as a

property of the fluid only.

The change in the law of resistance from the simple ratio has, however, been shown

by the author to be due to a change in the character of the motion of the fluid from

that of direct parallel motion to tha,t of sinuous or eddying motion, t

In the latter case, although the mean motion at any point taken over a sufficient

time is parallel to the pipe, it is made up of a succession of motions crossing the pipe

in different directions.

The question as to whether, in the case of sinuous motion, /x is to be considered as

a function of the velocity or not, depends on whether we regard f as expressing the

instantaneous shearing stress at a point, or the mean over a sufficient time. Whether

we regard the symbols in the equations of motion as expressing the instantaneous

motion or the mean taken over a sufficient time.

If the latter, then /x must be held to include, in addition to the mean stress, the

momentum per second parallel to ii carried by the cross streams in the negative

direction across the surface over which f is measured.

If, however, we regard the motion at each instant, then we must restrict our

definition of viscosity by making f the instantaneous value of the intensity of

resistance at a point.

This is a quantity which we have, and can have no means of measuring except

under circumstances which secure that /is constant for all points over a given surface,

and for all instants over a given time.

It thus appears that there are two essentially distinct viscosities in fluids. The one

a mechanical viscosity arising from the molar motion of the fluid, the other a physical

property of the fluid. It is worth while to point out that, although the conditions

under which the first of these, the mechanical viscosity, can exist depend primarily on

the physical viscosity, the actual magnitudes of these viscosities are independent, or

are only connected in a secondary manner. This is shown by a very striking but

little noticed fact. When the motion of the fluid is such that the resistance is as the

^ Reclaerclies Bxpl. Paris, 1852.

f *'An Experimental Investigation of the circumstances wHcli determine whether the Motion of

Water shall be Direct or Sinuous." Phil. Trans., vol. 174 (1883), p. 935.

AND ITS APPLICATION TO MR. B. TOWER'S EXPERIMENTS. 167

square of the velocity, the magnitude of this resistance is sensibly quite independent

of the character of the fluid in all respects except that of density. Thus, when in a

particular pipe the velocity of oil or treacle is sufficient for the resistance to vary as

the square of the velocity, the resistance is practically the same as it would be with

water at the same velocity, while the physical viscosity of water is more than one

hundred times less.

The answer, then, to the question as to the constancy of /x may be clearly given

—

jLc measures a physical property of the fluid which is independent of its motion. But

in this sense /x is the coefficient of instantaneous resistance to distortion at a point

moving with the fluid.

This restriction is equivalent to restricting the applications of the equations of

motion for a viscous fluid to the cases in which there are no eddies or sinuosities.

This, as shown by the author, is the case in parallel channels so long as the product

of the velocity, the width of the channel, and the density of the fluid divided by /x

is less than a certain constant value. In a round tube this constant is 1400, or

^<1400.

At a temperature of 50^, we have with a foot, as unit of length, for water :

—

^=0-00001428P

D'y<-02,

so that if D, the diameter of the channel, be *001 inch, v would have to be at least

240 feet per second for the resistance to vary other than as the velocity.

As regards the slipping at the boundaries, Poiseuille's experiments, as well as

those of the author, failed to show a trace of this, although / reached the value of

0*702 lb. per square inch, so that within this limit it may be taken as proved that there

is no slipping between any solid surface and water. With other fluids, such as

mercury in glass tubes, it is possible that the case may be different ; but, as regards

oils, the probability seems to be that the limit within which there is no shpping will

be much higher than with water.

12. Experimental determination of the value of fi for olive oil.

Since the value of /x for water is known for all moderate temperatures, in order to

obtain the value for oil it is only necessary to ascertain the relative times taken by the

same volumes of oil and water to flow through the same channel, care being taken to

make the channel such that there are no eddies and that the energy of motion is small

compared with the loss of head.

168 PROFESSOR O. REYNOLDS ON THE THEORY OP LUBRICATION

Tliese times are proportional to

Where p is the fall of pressure ; therefore the times multiplied by the respective falls

of pressure are proportional to the viscosities.

The arrangement of apparatus used is shown in fig. 2.

Pig. 2.

The test tube (A) containing the fluid to be tested v^^as fixed in a beaker of water,

which was heated and maintained at any required temperature from below,

A. syphon (B), made of glass tube i% inch internal diameter, with the extremity of

its short limb drawn down to capillary size for a length of about 6 inches, this six

inches being bent up and down so as only to occupy some 2 inches at the bottom of

the test tube. The long limb of the syphon extended to about 2 feet below the meanlevel of the fluid in the test tube. Two marks on the test tube at diifferent levels

served to show when a definite volume had been withdrawn.

The syphon used was the same for each set of experiments on oil and water, so that

the pressure urging the fluid through the tube was proportioned to the density of the

fluids—that is, it was 1*915 as great for oil as water, disregarding the efiect of the

variation of temperature on volume, which in no case amounted to 1 per cent.

Experiments were first made with water at different temperatures, the times taken

for the water to fall from the first mark to the second being carefully noted. Thesyphon was then dried and replaced and oil substituted for water.

Two sets of similar but entirely different apparatus were used on different occasions,

different samples of oil being used. In the first set the experiments on oil were made

AND ITS APPLICATION TO MR. B. TOWER'S BXPIRIMINTS. 169

at temperatures from 95^ to 200° Fahr. ; in the second set, from 61"^ to 120° Fahr.

In so far as the temperatures overlapped^ the viscosities for the two oils agreed to

within 4 per cent., bnt as the law of variation of the viscosity seemed to change

rapidly at about 140° Fahr., only the second set have been recorded. These are shown

in Table I.

From Poiseuille's experiments it is found that, measuring viscosity in pounds on

the square inch, for water at a temperature of 61° Fahr.,

^=10-"7xl-61.

Adopting this value of fi for the experiments on water at 61° Fahr., the other

experimental values of fi for water at different temperatures, obtained as being in the

ratios of the times, were found to be in very close agreement with those calculated

from Poiseuille's law for the respective temperatures. This tested the efficiency

of the apparatus. It has not been thought necessary to record any experiment on

water except at the temperature of 61° Fahr.

The experimental value of /x for oil are in the ratios of the times multiplied by

'915, the specific gravity of oil ; these are given as the experimental values of ft in

the table. Another column contains the values of /x for oil, calculated from an

empirical formula fitted to the experimental values._

This formula was found by comparing the logarithms of the experimental values

of ju,. It appeared that the differences in these logarithms were nearly proportional

to the differences in the corresponding temperatures, or that T being temperature in

degrees Fahr.,

in degrees Centigrade

whence since

logi^i-logf^3=-0096(T3-T0

log/^i-logM3='00535(T3-Ti)

•0096= -0221 logioe

•00535= -0123logioe

for degrees Fahrenheit

for degrees Centigrade/^3

/!l_g—0123(T,~T,)

(9\

This ratio holds well within the experimental accuracy from temperatures ranging

from 61° to 120° Fahr,, This is shown in the table, and again in fig. 3, in which

the ordinates are proportional to logf?,, the abscissae being proportional to the

correapojiding tepaperatures.

MDCCCLXXXVI. Z

170 PEOFESSOR O. REYNOLDS ON THE THEORY OF LUBRICATION

Table I.—^Viscosity of Oil compared with Water : 11 April, 1884.

JSTumber, Third.

Temperature,

Timeseconds.

107experimental.

log IX

experimental.logfx

calculated.

A*

107calculated.

Fahrenheit. Centigrade.

1

2

345

6

7

Water . .

» •

Olive oil .

11

o

60

11

11

618194

120

15-5

11

11

16

25

11

204013501000555

1-640

11

123-0081-00

60-00

33-40

5-089906-908485-778156-52375

5-0901336-898076-782906-52375

123-0679-08

59-3433-40

1%. 3.

13. The Comparative Values of //, for Different Fluids and Different

Systems of Units.

The values of /x given by different writers for air and water, are expressed in various

units of force and length, so that it is a matter of some trouble to compare them.

To facilitate this for the future comparative values are here given. Those for water

have been deduced from Poiseuille's formula, for air from Maxwell's formulae, and

for olive oil from the experiments recorded in the previous article.

The units of length, mass, and time, being respectively the centimetre, gramme^

and second, in which case the unit of force is the weight of one 980*5th (g) part

of a gramme, expressing temperature in degrees Centigrade by T and putting

p-i=:l+ 0-0336793T+0-0002209936T^ (3)


for

water . . /u,=0-017793lP

air . . . /i=0-0001879(l-|-0-00366T)

olive oil . . /A=3-2653 g-oiasT*

> (4)

With the same unit of length, but g grammes as unit of mass and 1 gramme as

unit of force, the values of /ia are for

water. . . /i,=0-000019792P

air .... /A=0-00000019153(1+ -00366T)

olive oil . . /A=0-0033303 e-oi^s^

«, • • • (5)

The units of length and mass being the foot and pound and the temperature in

degrees Fahr. for

water . . . /a=0-0011971P

air. . . . nt=0-00001l788(l+ -0020274T) }- .... (6)

olive oil . . )tt= 0-21943 e-022iT

With the same unit of length the unit of mass being 3'(32'1695) lbs. and the unit

of force 1 lb. for

water . . . /a=0-000037166P

air ... . /x=0-00000036645(l+ -00202741)

olive oil . . /x= 0-0068213 e-02211^ J-•< • . * (n

Taking the unit of length 1 inch and the unit of force 1 lb. for

water . . . /^=0-000000258105P

air . . . • /ji=0-0000000025447(l+'0020274T) }> . ... (8)

olive oil . . /x=0 •00004737 6-0^^1'^

* For olive oil tlie values of /* have only been tested between the limits of temperature lO"" and 49"^ C,

or 61*^ and 120° Fahr.

z 2

172 peo;fessoe o. reyi^olds on the theoey of lubrication

Section III.

—

General View of the Action of Lubrication.

14. The case of tivo nearly parallel Surfaces separated by a viscous Fluid,

Let AB and CD (fig. 4) be perpendicular sections of the surfaces, CD being of

limited but great extent compared with the distance h between the surfaces, both

surfaces being of unlimited length in a direction perpendicular to the paper.

Fig. 4.

Case 1. Parallel Surfaces in Relative Tangential Motion,—Li fig. 5 the surface CDis supposed fixed, while AB moves to the left with a velocity U.

Then by the definition of viscosity (Art. 9) there will be a tangential resistance

F= /^

U

and the tangential motion of the fluid will vary uniformly from U at AB to zero at

CD. Thus if FG (fig. 5) be taken to represent U, then PN will represent the velocity

in the fluid at P.

The slope of the line EG therefore may be taken to represent the force F, and the

direction of the tangential force on either surface is the same as if EG were in tension.

The sloping lines therefore represent the condition of motion and stress throughout

the film (fig. 5).

Fig. 5.

Case 2. Parallel Surfaces approaching with no Tangential Motion.—The fluid has

to be squeezed out between the surfaces, and since there is no motion at the surface,

the horizontal velocity outward will be greatest half-way between the surfaces,

nothing at O the middle of CD, and greatest at the ends.

If in a certain state of the motion (shown by dotted line, fig. 6) the space between

AB and CD be divided into 10 equal parts by vertical lines (fig. 6, dotted figure), and

these lines supposed to move with the fluid, they will shortly after assume the positions


of the curved lines (fig. 6), in which the areas included between each pair of curved

lines is the same as in the dotted figure. In this case, as in Case 1, the distance QPwill represent the motion at any point P, and the slope of the lines will represent the

tangential forces in the fluid as if the lines were stretched elastic strings. It is at

once seen from this that the fluid will be pulled towards the middle of CD by the

viscosity as though by the stretched elastic lines, and hence that the pressure will be

greatest at O and fall off towards the ends C and D, and would be approximately

represented by the curve at the top of the figure.

Fig. 6.

Case 3. Parallel Surfaces approaching with Tangential Motion.—The lines repre-

senting the motions in Oases 1 and 2 may be superimposed by adding the distances

PQ in fig. 6 to the distances PN in fig. 5.

The result will be as shown in fig. 7, in which the lines represent in the same way

as before the motions and stresses in the fluid where the surfaces are approaching with

tangential motion.Fig. 7.

In this case the distribution of pressure over CD is nearly the same as in Case 2,

and the mean tangential force will be the same as in Case 1. The distribution of the

friction over CD will, however, be different. This is shown by the inclination of the

curves at the points where they meet the surface. Thus on CD the slope is greater

on the left and less on the right, which shows that the friction will be greater on the

left and less on the right than in Case 1. On AB the slope is greater on the right

and less on the left, as is also the friction.

Case 4. Smfaces inclined with Tangential Movement only,—AB is in motion as in

Case 1, and CD is inclined as in fig. 8.

174 PBOFESSOR Q. REYNOLDS OK THE THEORY OF LUBRICATION

The effect in this case will be nearly the same as in the compound movement

(Case 3).

Eig. 8.

For if corresponding to the uniform movement U of AB, the velocity of the fluid

varied uniformly from the surface AB to CD, then the quantity carried across any

section PQ would beU

and consequently would be proportional to PQ ; but the quantities carried across all

Bections must be the same, as the surfaces do not change their relative distances

;

therefore there must be a general outflow from any vertical sections PQ, P'Q' given by

This outflow will take place to the right and left of the section of greatest pressure.

Let this be PiQi, then the flow past any other section PQ

f(PQ-PiQx)

to the right or left according as PQ is to the right or left of PiQi. Hence at this

section the motion will be one of uniform variation, and to the right and left the lines

showing the motion and friction will be nearly as in fig. f. This is shown in fig. 9.

Fig. 9.

This is the explanation of continuous lubrication.


The pressure of the intervening film of fluid would cause a force tending to separate

the surfaces.

The mean line or resultant of this force would act through some point 0.

This point O does not necessarily coincide with Pi the point of maximum pressure.

For equilibrium of the surface AB, will be in the line of the resultant external force

urging the surfaces together, otherwise the surface ACD would change its inclination*

The resultant pressure must also be equal to the resultant external force perpen-

dicular to AB (neglecting the obliquity of CD). If the surfaces were free to

approach the pressure would adjust itself to the load; for the nearer the surfaces the

greater would be the friction and consequent pressure for the same velocity, so that

the surfaces would approach until the pressure balanced the load.

As the distance between the surfaces diminished O would change its position, and

therefore, to prevent an alteration of inclination, the surface CD must be constrained

so that it could not turn round.

It IS to be noticed that continuous lubrication between plane surfaces can only take

place with continuous motion in one direction, which is the direction of continuous

inclination of the surfaces.

With reciprocating motion, in order that there may be continuous lubrication, the

surfaces must be other than plane.

15. Revolving Cylindrical Surface.

When the moving surface AB is cylindrical and revolving about its axis, the general

motion of the film will differ somewhat from what it is with flat surfaces.

Case 5. Revolving Motion, CD fiat and symmetrically placed.—1l}iq surface velocity

of AB may be expressed by U as before. The curves of motion found by the same

method as in the previous cases are shown in fig. 1 .

Wig. 10.

The curves to the right of GH, the shortest distance between the surfaces, will

have the same character as those in fig. 9 to the right of C, at which is also the shortest

distance between the surfaces.

176 PROFESSOR 0. REYNOLDS ON THE THEORY OF LUBRICATION

On the left of GH the curves will be exactly similar to those on the right, only-

drawn the other way about, so that they are concave towards a section at P^ in a

similar position on the left to that occupied by P^ on the right.

This is because a uniformly varying motion would carry a quantity of fluid pro-

portional to the thickness of the stratum from right to left, and thus while it would

carry more fluid through the sections towards the right than it would carry across

GH, necessitating an outward flow from the position Pj^ in both directions, the same

motion would carry more fluid away from sections towards C than it would supply

past GH, thus necessitating an inward flow towards the position P^.

Since G is in the middle of CD these two actions, though opposite, will be other-

wise symmetrical, andPgG= GPj.

From the convexity of the curves to the section at Pg it appears that this section

would be one of minimum pressure, just as P^ is of maximum. Of course this is

supposing the lubricant under sufficient pressure at C and D to allow of the pressure

falling. The curve of pressure would be similar to that at the top of fig. 10, in which

C and D are points of equal pressure, P^HP^ the singular points in the curve.

Under such conditions the fluid pressure acts to separate the surfaces on the right,

but as the pressure is negative on the left the surfaces will be drawn together. So

that the total effect will be to produce a turning moment on the surface AB.

Case 6. The same as Case 5, except that G is not in the middle of CD.—In

this case the curves of motion will be symmetrical on each side of H at equal

distances, as shown in fig. 11.

Fig. 11.

If C lies between H and P^ the pressure will be altogether positive, as shown by

the curve above fig. 11—that is, will tend to separate the surfaces.

AND ITS APPLICATION TO ME. B. TOWER'S EXPIRIMENTS. in

16. The Effect of a Limiting Supply of Lubricating Material.

In the cases already considered C and D have been the actual limits of the upper

surface. If the supply of lubricant is limited C and D may be the extreme points to

"which the separating film reaches on the upper surface^ which may be unlimited^ as in

fig. 12.

Eig. 12.

Case 7. Supply of Lubricant Limited.—1£ the surface AB be supposed to have

been covered with a film of oil, the oil adhering to the surface and moving with it,

then the surface CD to have been brought up to a less distance than that occupied

by the film of oil, the oil will accumulate as it is brought up by the motion of AB,

forming a pad between the surfaces particularly on the side D. .

The thickness of the film as it leaves the side C being reduced until the whole

surface AB is covered with a film of such thinness that as much leaves at C as is

brought up to D, then the condition will be steady.

Putting 6 for the thickness of the film of oil outside the pad, the quantity of oil

brought up to D by the motion of this film will be per second

and the quantity which passes the section PiQi, across which the velocity varies

uniformly, will bePiQiU

Therefore since there is no further accumulation

also, since GP2=GPi (fig. 10, Case 5)

MDCCCLXXXVI. 2 A


And since the quantity which passes P^Q^ will not be sufficient to occupy the larger

sections on the left, the fluid will not touch the upper surface to the left of Pg. The

limit will therefore be at P^, the fluid passing away with AB in a film of thickness h.

This is the ordinary case of partial lubrication : AB, the surface of the journal, is

covered with a film of oil ; CD, the surface of the brass or bearing, is separated from

AB by a pad of oil near H, the point of nearest approach.

This pad is under pressure, which is a maximum at P^,, and slopes away to nothing

at D and P|j,the extremities of the pad^ as- is shown l^ the curve above^ fig. 12.

1 7. The Relation hettveen Resistance, Ijoad^ amd Speed for Limited Luhrication,

In Case 7 a definite quantity of oil must be in the film round the journal, or in

the pad between the surfaces. As the surfaces approach the pad will increase and

the film diminish, and vice versd. The resistance increases with the length of the pad

and with the diminution of the distance between the sttrfe;ces.> The mean intensity

of pressure increases with the length of the pad, and inversely with the thickness of

the film, but not in either case in the simple ratio. The t^a| pressure which is equal

to the load increases with the intensity of pressure and the length of the pad.

The definite expressions of these relations depend on certain integrations, which

have not yet been effected. From the general relations pointed out, it follows that an

increase of load will diminish HG and P^Qi, and consequently the thickness of the

film round the journal^ and will increase the length of the pad. It will therefore

increase the friction.

Thus with a limited supply of oil the friction will increase with the load in some

ratio not precisely determined.

Further, both the friction and the pressure increase in the direct ratio of the speed,

provided the distance between the surfaces and the length of the pad remains con-

stant ; then, if the load remains constant, the thickness of the film must increase, and

the length of the pad diminish with the speed; and both these effects will diminish

friction in exactly the same ratio as the reduction of load diminishes friction.

Thus if with a speed U a load W and friction F a certain thickness of oil is

maintained, the same will be maintained with a speed MU, a load MW, and the

friction will be MF.How far this increase of friction is to be attributed to the increased velocity, and

how far to the increased load, is not yet shown in the theory for this case ; but, as

has been pointed out, if the load be altered from MW to W, the velocity remaining

the same, the friction will be altered from MF in the direction of F. Therefore, with •

the load constant, it does appear from the theory that the friction will not increase

as the first power of the velocity.

There is nothing therefore in this theory contrary to the experience that, with very

limited lubrication, the friction is proportional to the load and independent of the

AHD ITS APPLICATION" TO MR. B. TOWER'S EXPERIMENTS. 179

velocity, while the theoretical conclusioa that the friction, with any particular load

and speed, will depend on the supply of oil in the pad, is in strict accordance with

Mr. Tower's conclusion, and with the general disagreement of the coefficients of

friction determined in different experiments.

17a. The Conditions of Equilihrium with Cylindrical Surfaces.

So far CB has been considered as a flat surface, in which case the equilibrium of

CB requires that it should be so far constrained by external forces that it cannot

either change its direction or move horizontally.

When AB is a portion of a cylindrical surface, having its axis parallel to that of

AB, the only condition of constraint necessary for equilibrium is that CB shall not

turn about its axis. This will appear on consideration of the following cases :

—

Case 8. Surfaces Cylindrical and the Supply of Oil Limited.—Fig. 13 shows the

surfaces AB and CDPig. 13.

J is the axis of the journal AB.

I is the axis of the brass CD.

JL is the line in which the load acts.

O is the point in which JL meets AB.

R=:JR

Z TTfi

The condition for the equilibrium of I is that the resultant of the oil pressure on

BC together with friction shall be in the direction OL, and the magnitude of this

i^esultant shall be equal to the load.

2 A 2

180 PEOFESSOR O. REYNOLDS ON THE THEORY OF LUBRIOATIOiSr

As regards the magnitude of this resultant it increases as HG diminishes to a

certain limit, i.e., as the surfaces approach, so that in this respect equilibrium is

obviously secured, and it is only the direction of the resultant pressure and friction

that need be considered.

Since the fluid film is in equilibrium under the forces exerted by the two opposite

surfaces these forces must be equal and opposite, so that it is only necessary to

consider the forces exerted by AB on the fluid.

From what has been already seen in Cases 6 and 7 it appears that the resultant

line of pressure JM always lies on the right or on side of GH. The resultant

friction clearly acts to the left, so tlmt if JM be taken to represent the resultant

pressure and MN the resultant friction, N is to the left of M and JN the resultant

of pressure and friction is to the left of JM.

Taking LJ to represent the load, then LN will represent the resultant moving

force on CD that is on I. Since H wiJl move in the opposite direction to I, and

since the direction of the resultant pressure moves in the same direction as H, the eflect

of a moving force LN on I will be to move N towards L until they coincide^ Thus as

long as JM is within the arc covered by the brass a position of equilibrium is possible

and the equilibrium will be stable.

So far the condition of equilibrium shows that H will be on the left or off* side of

the line of load, and this holds whether the supply of oil is abundant or limited ; but

while with a very limited supply of oil, t.e., a very short oil pad, H must always be in

the immediate neighbourhood of O, this is by no means the case as the length of the

oil pad increases.

Case 9. Cylmdrical Surfaces in Oil Bath—If the supply of oil is sufficient the

oil film or pad between the surfaces will extend continuously from the extremities of

the brass, unless such extension would cause negative pressure which might lead to

discontinuity. In this case the conditions of equilibrium determine the position of H.

The conditions of equilibrium are as before—

1. That the horizontal component of the oil pressure on the brass shall balance the

horizontal component of the friction

;

2. That the vertical components of the pressure and friction shall balance the load.

Taking the surface of the brass, as is usual, to embrace nearly half the circumference

of the journal and, to commence with, supposing the brass to be unloaded, the move-

ment of H may be traced as the load increases.

When there is no load the conditions of equilibrium are satisfied if the position of

H is such that the vertical components of pressure and friction are each zero, and the

horizontal components are equal and opposite.

This will be when H is at (fig. 13); for then, as has been shown, Case 5, the

pressure on the left of H will be negative, and will be exactly equal to the pressure at

corresponding points on the right, so that the vertical components left and right balance

each other. On the other hand the horizontal component of the pressure to the left

AITD ITS APPLIOATION TO MR. B. TOWER'S EXPERIMEKTS. 181

and right will both act on the brass to the right, and as these wilF increase as the

surfaces approach, the distance JI must be exactly such that these components

balance the resultant friction, which by symmetry will be horizontal and acting to the

left.

It thus appears that when the brass is unloaded its point of nearest approach will

be its middle point. This position, together with the curves of pressure, are shown

in fig. 14.

Fig. 14.

As the load increases the positive vertical component on the right of GH must

overbalance the negative component on the left. This requires that H should be to

the left of O.

It is also necessary that the horizontal components of pressure and friction should

balance.

These two conditions determine the position of H and the value of JI.

As the load increases it appears from the exact equations (to be discussed in a

subsequent article) that OH reaches a maximum value, which places H nearly, but

not quite, at the left extremity of the brass, but leaves JI still small as compared

with GH.

For a further increase of the load H moves back again towards O.

In this condition the load has become so great that the friction which remains

nearly constant is so small by comparison that it may be neglected, and the condition

of equilibrium is that the horizontal component of the pressure is zero, and the

vertical component equal to the load.


H continues to recede as the load increases. But when HC becomes greater than

HP^y ^h.^ pressure between Pg and C would become negative if the condition did not

break down by discontinuity in the oil, which is sure to occur when the pressure falls

below that of zero^ and then the condition becomes the same as that with a limited

supply of oil.

This is important, as it shows that with extreme loads the oil bath comes to be

practically the same as that of a limited supply of oil, and hence that the extreme

load which the brass would carry would be the same in both cases—as Mr. Tower has

shown it to be.

In all Mr. Tower's experiments with the oil bath it appears that the conditions

were such tha.t H was in retreat as the load increased from towards 0, and that,

except in the extreme cases, Pg had not eome up to G.

Figs. 2, 3, 4 (Plate 8), show the exact curves of pressure as calculated by the exact

method to be given, for circumstances Gorrespoiiding very closely with one of Mr.

Tower's experiments, in which he actually measured the pressure of oil at three

points in the film. These measured pressures are shown by the crosses.

The result of the calculations for this experiment is to show, what could not indeed

be measured, that in Mr. Tower's experiment the difference in the radii of the brass

and journal at 70°, and a load of 100 lbs. per square inch,

a= -00077

GH= -000375

(The angle) OJH=48°

18. The Wear and Heating of Bearings.

Before the journal starts the effect of the load will have brought the brass into

contact with the journal at O. At starting the surfaces will be in contact, and the

initial friction will be between solid surfaces, causing some abrasion.

After motion commences the surfaces gradually separate as the velocity increases,

more particularly in the case of the oil bath, in which case at starting the friction will

be much the same as with a limited supply of oil.

As the speed increases according to the load, GH approaches, according to the

supply of oil, to a, and varies but slightly with any further increase of speed ; so that

the resistance becomes more nearly proportional to the speed and less affected by the

load.

When the condition of steady lubrication has been attained, if the surfaces are

completely separated by oil there should be no wear. But if there is wear, as it

appears from one cause or another there generally is, it would take place most rapidly

where the surfaces are nearest : that is, at GH on the off side of O.

Thus while the motion is in one direction the tendency to wear the surfaces to a fit

would be confined to the off side of O.


This appears to offer a verj simple and well-founded explanation of the important

and common circumstance that new surfaces do not behave so well as old ones ; and of

the circumstance, observed by Mr. Tower^ that in the case of the oil bath, running the

journal in one direction does not prepare the brass for carrying a load when the journal

is run in the opposite direction. This explanation, however, depends on the effect

of misfit in the journal and brass which has yet to be considered.

Case 10. Approximately cylindrical surfaces of limited length in the direction of 'the

axis of rotation,—l^oi\img has so far been said of any possible motion of the fluid

perpendicular to the direction of motion and parallel to the axis of the journal. It

having been assumed that the surfaces were truly cylindrical and of unlimited

length in direction of their axes, and in such case there would be no such flow.

But in practice brasses are necessarily of limited length, so that the oil can escape

from the ends of the brass. Such escape will obviously prevent the pressure of the

film of oil from reaching its full height for some distance from the ends of the brass

and cause it to fall to nothing at the extreme ends.

This was shown by Mr. Tower, who measured the pressure at several points along

the brass in the line through 0, and found it to follow a curve similar to that shown

in fig. 15, which corresponds to what might be expected from eilcape at the free ends.

f^^^- ^TTV^^,

o \>

C/t <sx^

>^ ^ 'S

V1

*̂*-„

N '

\ -s"'

V 1

\ s..

,v \,

V :\

I

^

iw

/

/

^J

\Jr ff / // r// I

. \

If the surfaces are not strictly parallel in the directions TU and VW, the pressure

would be greatest in the narrowest parts, causing axial flow from those into the broader

spaces. Hence, if the surfaces were considerably irregular, the lubricant would, by

184 PEOFESSOE O. EEYFOLDS OK THE THEOEY OF LFBEIOATION

escaping into broader spaces, allow the brass to approach and eventually to touch the

journal at the narrowest spaces, and this would be particularly the case near the ends.

As a matter of fact, the general fit of two new surfaces can only be approximate

;

and how near the approximation is, is a matter of the time and skill spent on preparing,

or, as it is called, bedding them. Such bedding as brasses are subject to would not

bring them to a condition in which the hills and hollows differed by less than a jo o o o^h

part of an inch, so that two such surfaces touching each other on the hills would have

spaces as great as a -g-^-Q^h of an inch between them. This seems a small matter, but

not when compared with the mean width of the interval between the brass and the

journal which, as will be subsequently shown, was less than x'ooo^l^ ^f ^^ inch.

It may be assumed, therefore, that such inequalities generally exist in the surfaces

of new brasses and journals. And as the surfaces according to their material and

manner of support yield to pressure the brass will close on the journal at its ends,

where, owing to the escape of oil, there is no pressure to keep them separate.

Fief, 16,'B

The section of a new brass and journal taken at GH will therefore be, if

sufficiently magnified, as shown in fig. 16, the thickness of the film, instead of

being, say, of Toooo^hs of an inch, varies from to joooo^hLS, and is less at the ends

than at the middle.

In this condition the wear will be at the points of contact, which will be in the

neighbourhood of GH on the off side of O (fig. 13), so tha.t if the journal runs in one

direction only the surfaces in the neighbourhood of GH (on the off side) will be

gradually worn to a fit, during which wear the friction will be great and attended

with heating, more or less, according to the rate of wear and the obstruction to the

escape of heat.

So long, however, as the journal runs in one direction only GH will be on one side

(the off side) of 0, and the wear will be altogether or mainly on this side, according

to the distance of H from O.

In the meantime the brass on the on side is not similarly worn, so that if the

motion of the journal is reversed and the point H transferred to the late on side the

wear will have to be gone through again.

That this is the true explanation is confirmed if, as seems from Mr. Tower's report,

the heating effect on first reversing the journal was much more evident in the case of

the oil bath.


For when the supply of oil is short HG will be very small and H will be close to 0>

So that the wearing area will probably extend to both sides of O, and thus the brass

be partially, if not altogether, prepared for running in the opposite direction.

When the supply of oil is complete, however, as has been shown, H is 50^ or 60°

from O, unless the load is in excess, so that the wear in the neighbourhood of H on

the one side of O would not extend to a point 100° or 120° over to the other side.

Even in the case of a perfectly smooth brass the running of the journal under a

sufficient load in one direction should, supposing some wear, according to the theory

render the brass less well able to carry the load when running in the opposite

direction. For, as has already appeared, the pressure between the journal and brass

depends on the radius of curvature of the brass on the on side being greater than

that of the journal. If then the effect of wear is to diminish the radius of the brass

on the off side, so that when the motion is reversed the radius of the new on side is

equal to or less than that of the journal, while the radius of the new off side is

greater, the oil pressure would not rise. And this is the effect of wear ; for as will

be definitely shown, the efiect of the oil pressure is to increase the radius of curvatiire

of the brass, and as the centre of wear is well on the off side, the effect of sufficient

wear -will be to bring the radius on this side, while the pressure is on, more nearly to

that of the journal, so that on the pressure being removed the brass on this side may

resume a radius even less than that of the journal.

Section IV.—The Equations of Hydrodynamics as Applied to Lubrication.

19. According to the usual method of expressing the stress in a viscous fluid (which

is the same as in an elastic solid) "^^^

:

F:£X'

Vyy

P~^'

p

V

„ (du , dv , dw\ . _ dib^

^^\dx dy ' dz

2(du dv dtd"

^^\dx dy dz 2f

P- 2 fdto dv diu\

dx

dv

^dy

dw

>^ (9)

pxi—''Pyx—l^\ ^^+ 7 )

'dvJ

d%i\

dx

Py^—PZlf

P^x—~-_Pxz

'dw dv

^^ dy dz,

dib dvj"

^^ dz da'.

> • • (10)

-/

* Store's '' On tlie Theories of tlie Internal Friction of Fluids in Motion, and of tlio Equilibrium and

Motion of Elastic Solids."—Trans. Cambridge Phil. Soc, vol. viii., p. 287. Also reprint, vol. i, p. 84.

Also Lamb's * Motion of Fluids,' p. 219.

MDCCCLXXXVT. 2 B

186 PROFESSOR O. REYNOLDS OIST THE THEORY OF LUBRICATION

In which the left-hand members are the stresses on plane perpendicular to the first

suffix in directions parallel to the second, the first three being tlie normal stresses, the

last six the tangential stresses.'

The values of these substituted in the equations of motion

6up-=pX+^+dy dz

hv

Pit

Phi-

hi'

pTx 4-dx dy

dy

dph

dz

'du dv diD"

'^dx dy dz

>

^

(10

give the complete equations of motion for the interior of a viscous fl.uid.

These equations involve terms severally depending on the inertia and the weight of

the fluid, also the variation of stress in the fluid.

In the case of lubrication the spaces between the solid surfaces are so small com-

pared with/^

U

that the motion of the fluid is shown to be free from eddies as already explained

(Art. 11). Also that the forces arising from weight and inertia are altogether small

compared with the stresses arising from viscosity.

The equations which result from the substitution from (9) and (10) in the first 3 of

(11) may therefore be simplified by the omission of the inertia and gravitation terms,

which are the terms involving /) as a factor.

In the case of oil the remaining terms may still further be simplified by omitting

the terms depending on the compressibility of the fluid.

Also if, as is the case, \x, is nearly constant the terms involving d^ may be omitted

or considered of secondary importance.

From equations (11) we then have

IdHi d^ub dhi

dx

dp

dy

dp

dz

0.

\dx^ dy' dz^

d^v

dy'^ dz^"^\dx?

Id^w d^w dhv

''^\d^'^dy^'^d^

du dv dw

dx dy dz

r (12)


Again, since in the case of lubrication we always have to do with a film of fluid

between nearly parallel surfaces, of which the radii of curvature are large compared

with the thickness of the film, we may without error disregard any curvature there

may be in the surfaces, and put

X for distances measured on one of the surfaces in the direction of relative

motion,

% for distances measured on the same surface in the direction perpendicular to

relative motion,

y for distances measured everywhere at right angles to the surface.

Then, if the surfaces remain in their original direction., since they are nearly parallel,

V will be small compared with u and iv^ and the variations of u and w in the

directions x and % are small compared with their variations in the

direction y.

The equations (12) for the interior of the film then become

Equations (10) become

d'p d?u~>i

dx

dp

dy

dz

0:

^d'lf'

^

d^w

df7^ ^,,3

d%b dv dw

dx dy dz

• • (13)

^

Pxy— pyx— /^

Pyz—Psy— /^

_PzX P-'T-^ ^

d'li~1

dy

dw

dy

. (14)

J

20. The fluid is subject to boundary conditions as regards pressure and velocity.

These are

—

(1) At the lubricated surikces the fluid has the velocity of those surfaces;

(2) At the extremities of the surfaces or film the pressure depends on external

conditions.

Thus taking the solid surfaces as 2/=0, 3/= A, and as being limited in the direction

X and z by the curve

2 B 2

188. PR0FE3SS0B 0. BETFOLDS OF THE THEORr Of LUBRICATIOF

For boondarj conditions

y u Uq vj v—0 1

y-h to Uj w — •'-^.1!+^.

f{xy)— Q p==Po

• « (15)

21. Equations (13) may now be integrated, the constants being determined by the

conditions (15).

The second of these equations gives p independent of y^ so that the first and third

are directly integrable, whence

1 clr.

U'.

IV-

iyju jjj Li/Ju

1 dp,

ilii ciz

K)y+^^-^+yih>

h)y

• « * • • (16)

Differentiating these equations with respect to x and z respectively, and substituting

in the last of equations (13)

dv

dy'

1^

2lJL'^%-Â+i\i{y-i-)y

Cl"A/I Ci/iA/ dz I dz i[^'^+K

Integrating from 2/=0 to y=/^, and substituting from conditions (15)

£('''l)+l("f)=«''{(^'»+u.)S+^^. . . . . (17)

From equations (16) and (14)

dpp^-=-^jpy^^)-^tîî

.dp ,

^y—h)

» « » 9 3 , (18)

Putting ^jC for the shearing stresses at the solid on the surfaces in the directions

X and z respectively, then taking the positive sign when y=^h, and the negative

when y=:0

/.=/.(U,-Uo)^=Fi|A

/,=:= rpiJ/i

r « « $ « ft 9 t (19)

J

AND ITS APPLICATION TO MR. B. TOWEE'S EXPERIMENTS. 189

Equations 17 and 19 are the general equations of equilibrium for the lubricant

between continuous surfaces at a distance h, where h is any continuous function of

X and z, and /x, is constant.

22. For the further integration of these equations it is necessary to know the exact

manner in which x and z enter into h, as well as the function which determines the

limit of lubricated surfaces.

These integrations have been effected either completely or approximately for certain

cases, which include the chief case of practical lubrication.

Complete integration has been obtained for the case of two parallel circular or ellip-

tical surfaces approaching without tangential motion. This case is interesting from

the experiment, treated approximately by Stefan,^ of one surface-plate floating on

another in virtue of the separating film of air. It is introduced here, however, as

being the most complete as well as the simplest case in which to consider the

important effect of normal motion in the action of lubricants. It corresponds with

Case 2, section III.

Complete integration is also obtained for two plane surfaces

between the limits at whicli p=Il (the pressure of the atmosphere)

,//— vj, X——(jjy

the surfaces being unlimited in the direction of z. This corresponds with Case 4,

section III.

For the most important case, that of cylindrical surfi;ices, approximate integration

has been effected for the case of complete lubrication with the surfaces unlimited in

the direction of z. Case 9, section III.

Section V.— Cases in which the Equations ahe Completely Integrated.

23, Two Parallel Plane Surfaces approaching each other, the Surfaces having

Elliptical Boundaries,

Here h is constant over the surfaces, and when

x^ z^

I j»—- j_ // 11 I . X.JL t t » . . » . \ ^ V/ /

a^ ' c

Uq, Ui are izero.

Equation (17) becomes

[dx'^dzyP^ ¥ dt ' ' ^' ^^^

The solution of which is

x^^ . z^

jp— (^(j^)^^-f^-f C^}>+Eie'"^sin^+&c. . , . . , , (22)

a"" c

^' Wien, Sitz, Ber,, voL 69 (1874), p. 713.

190 PBOFESSOE O. EBrFOLDS ON THE THEORY OF LUBRICATION

Therefore

2<^(0f:^+i)='lf? • • • (23)a' W dt

and

Ei=0, &c.

C,

p-U

1+n

</,(0

12[jb a^c^r

h^ a^+ <?^

3

a^ c^1

dh>dt

(24)

From equations (19)

/.

f.^

24iCfc a^c^ dh1— .j^ fy*

h^ a?+ c^ dt

_^ 24iCfc a^ dhI „

i—. 4 ly

h^ o?+ (? dt

'^

• (25)

supposing surfaces horizonta] and the npper surface supported solely by the pressure

of the fluid. The conditions of equilibrium in this case are obvious by symmetry.

The centre of gravity of the load must be vertically over the centre of the ellipse.

Since by symmetry

^fV(-f;) pxdxdz=^0

re Pa/(i--J)"^ pzclxcl

is/i^-t)

o-'o ^

(26)

And

W:r\/i-

0-^

p -— lidxdy

S/jLft a^c^ dh

K^ d^+ c^ dt

(27)

(28)

Therefore intepfrating"

t=SfjbTraV ( 1

(a^+ cJW\\ K^(29)

t being the time occupied in falling from h^ to Ag.

24. Plane Surfaces of unlimited Length and parallel in the Dirrxtion ofz.

The lower surface unlimited in the direction x and moving with a velocity — U.

The upper siuface fixed and extending from x=^0 to x^=a. This case corresponds

with Case 4, Section III.

AND ITS APPLICATION TO ME. B. TOWER'S EXPERIMENTS. 191

The houudary conditions are

x=0

x=ap:==U

a.

^

y=:0 Uo=-Uy=h Ui= Vi=0

J9 is a function of x only.

And from equation (17), Section, IV,, by integration

^ * • » . . . . (30)

—

—

U^ \J - „ • f . f f t « t » » lOi/cvx M

hj being the value of h wben m=Xi where the pressure is a maximum.Integrating with respect to x, and puttingp=n at the boundaries

X

^ ,aVa r 1

a^'^2 +m ' • !

1 + m 1

« •¥ • t t » »

1 "1

V,

^ ,SO 2-\-m L. X

1 + m- 1 4 m-a \ a,

2+mJ

(32)

» » (33)

Ca

Q

(^•-n)rf^=~fi:;:^ ^ log, (14-^)—— v

hhii^1 +

771

V,

• t 1 '. . . (34)

^

or putting W for the load per unit of breadth, W is a maximumapproximately and

w=l*2

again, by equation (19)

therefore

and if

W .a'

ajL \J Im Vy ^ r% * % • •

I ^3

:'_..Ul

? « » » • (35)

/^=^A-

? « f « t •<! « t e (3G)

a

f4x-=:~- log, (1+m)

m=:r2

< « ? • • • (37)

F=-6572A,

a * ij > * < i . , (38)

192 PROFESSOR 0. REYNOLDS OX THE THEORY OP LUBRICiiTIOisr

In order to render the applicat'on of equations (35) and (38) clear, a particular case

maj be assumed.

which is the value for olive oil at a temperature of 70° Fahr.^ the unit of length being

the inch, and that of force the lb.

LetU= 60 (inches per sec.)

^^='0003.

Then fi*om (35), the load in lbs. per square inch of lubricated surface is given by

W:1070(X^

a

and from 38, the frictional resistance in lbs, per square inch is

F~:=1-31a

This seems to be about the extreme case of perfect lubrication between plane metal

surfaces having what appears to be about the minimum value of h^.

SECTioisr VI.

—

The Integration op the Equations for the Case op

Cylindrical Surfaces.

25. General Adaptation of the Equations,

Fig. 17 represents a sectioii of two circular cylindrical surfaces at right angles to

the axes ; as in Art. 1

7

J is the axis of the journal AB

;

I is the axis of the brass CD ;

JO is the line of action of the load cutting the brass symmetrically, and

R=JPR+a=IQ

h=VQA^=:HG, the smallest section.

I

Jl=zGa y * ' ' (39)i

!

GJ0=:|-4

P-^JO=<^i. Pj being the point of maximum pressure. J

AND ITS APPLICATION TO MB. B. TOWER^S EXPERIMENTS, 193

Fig. 17.

Then taking x for distances measured in the direction OA from on the surface

AB, and putting t for the distance of any point from J

(40)

î=R</>i (41)

Neglecting quantities of the orderca

E

3

/2.=a{l+c sin {d—ô}} (42)

For if I be moved up to J, Q moves through a distance ca in the direction JH.

The boundary conditions are such that

(1) all quantities are independent of z

(2) Uo is constant, Uj and V]^=0

(3) putting ^Q=OJA, 1=^0JIB, v^hence by symmetry ^Q=r—-^^

MDCCCLXXXVI.

e=eA

2 c

P=2ô (43)

194 PROFESSOR 0. REYNOLDS 01^- THE THEORY OF LUBRIOATIOISr

Putting —L for the effect of the external load and —M for the external momentper unit of length in the direction z^ and assuming that there are no external

horizontal forces, the conditions of equilibrium for the brass are

•^1

YP sin O—fco^ 0}d0=OOo

m

\{pcose+fsmd}de=^ (45)

rfde=M

(46)

Substituting from equations (40) (41) (42) in equations (17) and (19), Section IV.,

putting

K,=^55., K,=^^-^ . (47)

and remembering the boundary conditions, these equations become on integration

dp ^Rfi\J^c{ sin {0—^^— sin (0i— ô)}

a2{l + csin((9-ô)}'

3//,Uoc{ sin((9-</)o)- sin {(jy^-^^)}

<x{H-csin(6— ^)}'

//,Uo

{H-csin(^— ^)}

(48)

(49)

26. The Method of Approximate Integration.

The second numbers of equations (48) and (49) may be expanded so that

1 dp

k"c^^"^^"^0+^1 sin (^-</>o)+A3 cos 2(^-(^)+, &c.

+A3;, cos 2a^(<9— (j?>o)+A2,^+isin {(2^+l)(<9-~^)}

> . . . . (50)

J

K

Putting

-/=Bo+Bi sin (^-(/>q)+B3 cos 2(^-</>)+ &c.

+ 63^ cos 2cc(^-<^)+B2«+i sin {(2a;+l)(^-<^)}

X=sm(<î-<ô)

• • • • (51)

(52)


r=2oo

^0 — "~X""^.,-9-^

(r4-l)r^(r— 1) . . . --^^ (7' -{- 2)(r+ iy(r— 1) . . . —^•^

2

T-22r+l 2»'+i

c'-x )-

XXn^ —

—

(""" X I

"^

{2n+ l)2n „ .,,(2ji4-2)(2ti+l)

23«c^^^+i-f ,%n

2%n c-'x

r=2oo

* r=2n+2

(r-hl)r\r—l)r+ 2n + 2

(r4-2)(r+ l)r . . .

r+ 2n+ 2~

O^—l-j^

^r c-x (53)

2^+1— ^— j.^ ^(27i+2)(2%+l) (2w + 3)(2?i+2)

g^^j22^+1

'C^+ 22W+1 C--X

r=2oo + l

r=2n+d

(r+l)r^(r-— 1) . . .

r+2^4-3(r+ 2)(r+ l)r . . . r-h2n+ 2

r—2n—lCT J. . I

I

22.r— 2?z-—

1

c^X

2

>

J

Bq =1—3cx4-2r=2oo

9«=2

(4-3(r+ l))n(r-l)...'r+22

2^

2

3(r+l)f(r— 1) . . .

^'+ 1

2

2''T

2

.^.-M^>

J

-N

X^o^ ——" \^ X I ^

[4 - 3(27t+ l)]c^''- 3(2n + l)e^''+ix

""J^** jlLj

r=2oc

r=2^t+2[4-3(r+l)]c'--3(r+iy"x

^.(r— 1) . . .

T+ 2n+ 2'\

2 I

f-i r

—

2n2

(54)

i^3?2+i— ( 1/ '\

[4_ 3(2^ 4- 2)]c3^+i- 3(2^+ 2)c2^+3;j^

23»

r=2oo+ 3

-L.Vr=2n+3

[4-3(r+l)>/-3(r+l)c'-+ixr.(r+ l) . . . T-^2n-\-Z

2^-1r— 2^1—

1

2

2 c 2

169 PROFESSOE 0. RBYFOLBS OK THE THEORY OP LITBRICATIOK

Tlie coefficients Aq^ A^, &c., Bq, B^, &c., are thus expanded in a series of ascending

powers of c with numerical coefficients w^hich do not converge. It seems, however,

that if c is not greater than '6 the series are themselves convergent, and it is only

necessary to go to the tenth or twelfth term, to which extent they have been

calculated, and are as follows :—

A0'

XJL.|

XXO

J\.i

-l-5c-3-75c3-6-565c5-9-8r)c''-13-51c5'-l7-6cii

-{l+ 3c2+5-625cH8-75c«+12-225cS+16-2cW}^

l+ 4-5c3+9-375c4+15-23c«+21-92c8+29-8cio+38-6ci2

+ {3c+7-5cHl3-13c^-i-19'7c7+27-0lc9+35-2c"}x

l-5c+5c3+9'85c»+15-75cM-22'56c9+20-24cii

+ {3cH7;5c*4-13'13c«+197c8+27-02cio+35-2ci2}x

— l'5c^— 4-7c*— 9-2c«— 14-7oS— 21-45cô

-{2-5c^+6-56c^+ll-78c7+18'03c9+25'4c"}X

A4=~l-25c3-3-94c5— 7-875c7— 12-88c''-18-8cii

-{l-875c*+5-25c«+9-85c8+15-48ci04-22-45c^3j^

A5=-939c*+3-07c8+6-33c8+10-68cio

+ {2-63c6+3-94c7+7-76c9+]2-6cî}X

y . (55)

-^

Bo=l- {2-5cH4-125c*+5-3125c«+6-54c8} 1

— {3c+4'5c3+5-625c5-|-6-562c7+7'63c»}x

Bi=2c+6c3+9-75c5+ 11 -312507

+ {6cH9c*+ll'25c8+12-5c8]x

B2=2-5c2+5-5cH''"97c«+10-06c8

T>-D3=

B.

+ {4-5c3+7-5c5+9'85cHll"8c''}x

2c3— 4-375c5— 6-5625C''— 8-61c^

-{3eH5'(J25c8+7-873c«+9-84cô}x

1-375C*— 3-2c8-5-03c8

- {l-875c^+3-94c7+5'09c''}x

(56)


27. The Integration of the Equations,

Integrating equation (50) between the limits 0^ and 6

Ue-d,)

— Ai{cos {0—(}>q)— eos (6'o— <ô)}

+f {sin 2{0-cj>o)- sin 2(ô-'ô)}

&c. &c.

27^+1{cos [(2ti+l)(^-(ô)]- cos [(2b+1)(ô-<ô)3}

+|j{sin2n(^^(ô)-^m2^ô-<ô)} (57)

whence putting 0=:d^hj condition (43)

A .0=Aqî—Aj sin $1 sin <ô+'o^ ^^^ ^^^^ cos 2^q— , &c.

+Tr^ sin 2nî COS 2^^Q (58)

Putting

A'Ez=Ai cos ^1 COS ^0+^ COS 2^3^ sin 2<^q+5 &c.

A+-^ cos 2nî sin 2n^Q ........... (59)

whence from equations (57) and (58)

^^«=E+Ao^-Ai cos (^-(ô)+4' î^ 2(^--.<ô)

A ,

+~^sin2^^— ^q) . ........... (60)


Multiplying equation (60) by sin 9 and integrating between 9q and 9^, remember-

ing that 9q-=-—9i

f

'^—^sin 9d9= 2A,{sm 9^-^ 9^ cos 9,)

J Oq I^<^

, . /sin 26. sin 4>o n -t+ Ai( 1 ^'-^1 sm (j>^

, Ao/ . ^ ^ , sin 3^ cos 26'

+Y sm l9 cos 2</> r^

&c. &c.

+A 2«+l

27^+ l

sin (2^^+ 2)^J sin (27^+ l)(ô î^ ^^z.^^ sin (2?^+ l)(^(,

27^+2""

2?z,

Ag^ Jsin (2??.+ 1)6^ cos 27^^ sin (27^ + l)^ cos 2n^

2n\ 271-1""

^ 2n+

1

. . . . (61)

Multiplying equation (60) by cos 9, and integrating from ^q to 9^

f

^'V-'Vn ^7^ . /^ ,sin 2^1 I \ .

A„ /sin 3^1 . , « . ^ . v

kT ^^^ ^ — Ai(6>i cos(/>o Y^ cos <ô)+ -^(" g— sm 2^0—f sm l9sm 2^)— &c.

A o;2+-l

27^-f 1

'2n-\-l 2n-\-lY^ dii{2n+2)9 cos (2n+ 1)^ 272" ^^^ ^^^^ ^^^ ^^^^+ ^^^

+A

2^2

27^

272, 272-

^—Ysin(27^+l)^sin27^^——— sin(27^— ]^ l. . (62)

Multiplying equation (51) by cos 9 and integrating

e fcosO o-D •zi -o

/sin 2(9^ sinô,^ . .

do Kg

, -r, /sin 3^. cos 26r. . ^+BJ ^- ^^+ sm ^1 cos2(/>o

&c.

+ B3«

sin (27^ +1)6^1 cos 27^^Q sin(27^— 1)6^^ cos 27z.ô

27^ + l+

27^-l

JL>Om_L.T "

2«+l

sin (27Z+ 2)^1 sin (27Z>+ l)<ô sin 2?20i sin (272+ l)<ô

27^ + 2 271 } • . (63)

AND ITS APPLICATION TO MR. B. TOWER'S EXPERIMENTS, 199

Multiplying equation (51) by sin 9 and integrating

r/sin 6dd_^ /sin 26^ cos ^p ^ ,_- — i3il

^^

~t)^ cos ^0-r. /sin 3^, sin 2c6o . /i . ^^ i \

BB«

sin (2^+ 1)^1 sin 2n^Q sin (29^— 1)6^ sin 2^?.^q

2^t.+ l 2^— 1

+B *sin (2b 4- 2)^^ cos (2b4-1)ô ^^^ ^^î ^^^ (2%— 1)(^

P+i 2^2,4-2 2'?^1 . . . (64)

Integrating equation (51)

[îfdd

J 00 3

0"R

= 2B(j^jL"~"2Bi ^1^ ^1 si^ ^0+ "IT ®^^ ^^1 ^^® 2^0

2B+^sin2n^,cos2#o

2^2.

2B2^+1

2n+ lsin (2ti+ 1)^1 sin (2^+1)^0 ........... (65)

Substituting from equations (61 )~(6 5) in the equations of equilibrium (44), (45),

(46), there results-

From (44)

2(KicAq+K2Bq) sin ^j— 2KicAq^| cos 9i

+ (KioAj— KgBj) —^--^ sin <^0— (KicA^+K^Bi)^! sin ^,

+ &c.

^=00

+ 2^=1 *

^K|CA|Î TT "o \ sin (2b—-1)^^ cos 2^^^

'KjcAg^ \ sin (2^+ l)^j cos 2%^^

JL\. » C/xl.2»+l

2#+l. ,I.\ ,0-LJi

sin 2n6i sin (2^+ l)<ô

sin {2o%+ 1)^1 sin {2n+ 1)^^2-^2^?+l 2n+ 2

« *

From (45)

LIt

(KicAi-K2Bi)(^^\^-î ) cos <ô

.^t=oo

-T"Sî=i "! \îi.|CA2« -r i^^glig^j

I (-'^l^-^2?*+l J^S-I^S^+l)

sin (2^ 4- 1)^1 sin (2^2. --l)^^"

2n-^l 2#—

1

sin {2n+ 2) ^;isin 2n0-l

2^^.4-2 2n

(66)

sin 2iiô

cos(2n+l)<ôl • (6^)

200 PEOFESSOR 0. REYNOLDS ON" THE THEORY OF LUBRICATION"

rom (46)

2B2Bq^j— 2Bi sin 0^ sin ^qH—^ sin 2^^ cos 2^q

2

2^

2Bc

4--;7^ sin 27^^J^ cos 27^^Q

_r_l!!±lg|j2 ^2ti+ 1)^1 sin (27^+l)<ô r (^^)

The equations (66), (67), (68), together with (58), which, expresses the boundary-

conditions as regards pressure, are the integral equations of equilibrium for the fluid

between the brass and journal, and hence for the brass.

The quantities involved in these equations are

R, U, M, L, /x, 0^ and a, c,(fy^, (^^

If, therefore, the former are given, the latter are determined by the solution of these

equations.

Section VII.

—

Solution of the Equations for Cylindrical Surfaces.

28. c and A/ ^ small compared with Unity.

In this case equations (55) become

Ao= -"X Bo= 1

Ai= 1 Bi=0 }> • (69)

A^= 0, &c. B^— 0, &c.

Equation (58) gives

Equation (66) gives

0=;)(^2+ sin ^1 sin ^Q (70)

= {2KiCx—2K^) sin ^j—•2KjC;yî ^^^ î— K^c —~-~^ sin ^q+KiC^*! sin ^q . (71)

Equation (67) gives

L ^^ /sin 2d~t^^ /sin 2^, ^ \, .^.= KiCi—r—^—

•fc'ijcos ^0 ........ (72)

and equation (68) gives

M

Also equation (57)

i^—Po=Kic{cos îcosô—X^— cos (^— ^)} (74)

AND ITS APPLICATION TO MR. B, TOWER'S EXPERIMENTS. 201

Eliminating x between equations (70) and (71)

sin ^0=K, 2 sin ^1

î Jd-^^ + ism20(75)

The equations (74) and (75) suffice to determine a, c, î and ^q under the conditions

a/» and c »,naU so long a. .^. k not small, in which case the terms retained in the

equations become so small that some that have been neglected rise into relative

importance.

To commence with let

(Cases 5 and 9, Section III.)

Then by (72)

L=0

and by (70)

putting for ^ its value sin (^^— ^q)

cos^Q=0

sin 9^

X=

COS<^j

01

sin 0^(76)

Equation (76) gives two equal values of opposite signs for (j)^. These correspond to

the positions of Pj^ and P^, the points of maximum and minimum pressure.

For the extreme cases

^1=0 </.i=±vl^,^

TT

I

IT

î=±:COS9,

TT

-/


a=:

and from (75)

M

a sin 6.

When L increases

From (72) and (75)

3E /a 2 sin^ ^^ ,, . o/i \

•

(77)

• ('8)

• « (79)

R^r- /sin 20^ ^\ 2 sin 0jtan *,=jK,(^ -»,j^--^^---y . .

Hence as L increases tan cjj^ diminishes until the approximation fails,

ever, does not happen so long as c is sraall.

MDCCCLXXXVI. 2 D

. . (80)

This, how-

202 PBOFESSOR O. EEYISrOLDS OK THE THEORY OF LUBRIOATIOJST

As the load increases from zero, equation (80) shovfs that G moves away from Otowards A.

It also appears from equation (70) that x ^^d ^^ diminish as <ô diminishes, and that

(f>iis positive as long as the equations hold.

To proceed further it is necessaiy to retain all terms of the first order of small

quantities.

Ketaining the first power of c only, equations (55) become

Ao=~l-5c-~x Bo-l-3ox -^

A,- 1+3CX Bi 2c I

A^— l'5c B.-0 J

From (58)

x(î+3c sin 0^ sin ^q)"~ """ ^'^^

î sin iô' l-5o^,+ l-5c"'"f^ cos2(f>Q

From (66)

From (67)

From (68)

MR^

—K3{2(1— 3cx)î"~"^^ ^11^ ^1 Q^'^ ôl • *

(81)

(82)

. . . (83)

0={-2Kic(l-5c+x) +2X3(1 -3cx)} sinî

+ 2Kjc(l-5c+x)<î cos ^1

+ {Kic(l+ 3cx)— 2K2C} '^^^^ sin (j)Q

— {Kic(l+ 3cx)+ 2K2c}î sin <ô . . . •

+fKic3(sinî-^^^ cos 2,^0

-= -{KiC(l+3cx)-2K3c}fî—^-^^^j cos<ô

+KiC—-"(i sin 3^1 sin 2<^q— sin 0^ sin 2<^q) . (84)

» • • . (85)

In the equations (82) to (85) teimis have been retained as far as the second

power of c, but these terms have very unequal values. As x ^l^d sin <^q diminish c

increases, and the products of ex or c sin <^q may be regarded as never becoming

importa-nt and be omitted when multiplied by K^c or K^.

Making such omissions and eliminating x between (82) and (83)

sin (ji0'

0^ sin 6^ + c msm0^'-'sin 30.

oO

SCsinî-îCos^Z-H';^!

/^ sin 2Â^1 ^1-—

T~ -2(sin^, 0^ cos 0i) sin 0^

(86)


Equation (86) is a quadratic for c in terms of sin <^q, from which it is clear that as c

increases from zero <^q goes through a minimum value when

K, ./. /, sin3(9,\ , ^ ^ ^ sin 2^, • • • •V^^/

"^J |îfsm(9i 3-^j-~3(sin6'i-~6>iCos6'i)--Y--^

As the load increases from zero the value of c increases from that of equation (79)

to the positive root of (87). As the load continues to increase c further increases, but

<^Q again increases, so that, as shown by equation (86), for values of <^q greater than the

minimum there are two loads, two values of c, and two values of x-

If 0^ is nearly ~ c will be of the order a/^ when <^q is small, and sin ^q will be of

the order Ac ; so that, so long as ,\/~ is sufficiently small, no error has been intro-

duced by the neglect of products and squares of these quantities.

For example

(9^= 1-87045 (78'' 31' 30" as in Tower's experiments) .... (88)

By equation (86)

And by (87) at the minimum value of ^q

asin<ô=3-934c+l-9847-p . (89)

= 'V/a

2R

a(90)

sin ^0=5*61 /\/p^

Putting y= sin <î—• sin<j&o

eqâ^tion (82) becomes

sin <î== --167760+ -56561^. . . . . . . . (91)

or, when <j&q is a minimum,

sin (j!>i= '682^1 . (92)

Therefore _

X=4-928 ^Jl (93)

Equations (84) and (85) give

E

'zr^ '^^—^ —— 2i * i 4IVo « • • » * • • • • \* /

2 D 2

204 PBOFBSSOR 0. REfFOLDS ON THE THEORY OF LUBRICATION

whence equations (47)

Lc=:'388a--z ........... (96)

a K^Uo ^ ^

LSo long then as airj is not greater than 0*2, these approximate solutions are

sufficiently applicable to any case.

For greater values of rj the solution becomes more difficult, as long hov/ever as c is

not greater than '5 the solution can be obtained for any particular value of c.

29. Further Approximation to the Solution of the Equations for particular

Values of c.

The process here adopted is to assume a value for c. From equations (53) and (54)

to find

Ao=Ao'+Ao"x Bo=Bo'4-Bo"x. •..... (98)

&c. &a

where A/ A/^ B/ B/^ are numerical.

These coefficients are then introduced into equations (58) and (66) which on

eliminating x give one equation for ^q.

The complex manner in which(f>Q

enters into the equation renders solution difficult

except by trial, in which way values of ^q corresponding to different values of c have

been found.

The value of <^o substituted in equations (58) or (66) gives x ^^d ^pThe corresponding values of c^,

(fy^said x being thus obtained, a com^plete table might

be calculated. This, however, has not been done, as there does not exist sufficient

experimental data to render such a table necessary.

What has been done is to obtain(f)^

and(J)q

for c=:'5, Oi having the value 1'3704 as

in equation (88) and in Tower's experiments.

The value c^=^'5 was chosen by a process of trial in order to correspond with the

experiments in which Mr. Tower measured the pressure at different parts of the

journal as described, in his second report, and as being the greatest value of c for

A^'hich complete lubrication is certain.

Putting c='5, equations (43) and (44) give


Aq -1-5351 -2-3018X Jô

Ax 3-0723+ 3-0721X • AA, 1-86474-1-5360X B,

A3—- -sgii— -esTix JB3

A, -- -3753— •2582x B,

A5- •1396+ •1343x

- -012— 2-304x

2-143+2-286X

1-139+ -saex

- -455- •316x

- -146— •097x

(99)

lakmg '^

^,= 1-3704 or 78° 31' 30

it was found by trial that when

<ô=48'

(100)

-/

and K^ was neglected (under the circumstances Kg was about 'OOOSKi), equation (58)

gave

X= — -82295 or -^ sin 55° 22' 40""

and equation (66) gave

X=— -82274 „ -^ sin 55"^ 21' 40'

>..... (101)

The difference '00021 being in the same direction and about the magnitude of the

effect of neglecting K^.

This solution was therefore sufficientlj accurate, and adopting the vahie of ^1—^0

î=:7° 2r 40".• • • * • • ».• !* ^ *^ /

Equations (99) then became

Aq— -3587 Bo= 1-9 12

Ai= -5449 Bx= 2-263

A3— -eoio 63= •303

A3=--3505, B3—- •195

A4=— -0407 B,= •066

A5----O29I,

. . . . (103)

Substituting the values from equations (100), (102), and (103) in

equation (67) Xj— —^X'JiJqJjjSujC • » S • "9 9 «'# Jl'-S(

equation (68) 4'7546K3 (105)

206 PROFESSOR 6. RBYlSrOLDS OX THE THEORY OF LUBRIOATIOK

By equation (59)

By equation (60)

E=— -25257 (106)

— •25257H--3587^— -545 cos (6'— 48°)

+ -3005 sin 2(^—48°)+ -1168 cos 3(^—48°)

— -0407 sin 4(6'— 48°)+ -0058 cos 5((9 -48°) (107)

From equation (107) values of

*^

have been found for values of 6 diflfering by 10°, and at certain particular values

of^-^=±29° 20' 20^' points at which the pressure was measured

6=^ — 7° 21' 40'' point of maximum pressure

6=^ — 76° 38' 40" point of minimum pressure

0z=z-^7%^ 31' 30" the extremities of the brass

> (108)

These are given in Table II.

Table IL—The pressure at various points round the bearing.

S oflf side. Arc radius =1. P -Pa6 on side. Arc radius = 1.

P-Po

o / //

0-0 1-0151o / //

1-0151- 7 21 40 -0-12837 1-0269-10 -0-17453 1-0232 10 0-17453 0-9331-20 -0-34907 0-9412 20 0-34907 0-8022-29 20 20 -0-5120 0-7923 29 20 20 0-5120 0-6609-40 -0-6981 0-5612 40 0-6981 0-5003-50 -0-8737 0-8349 50 0-8737 0-3555-60 -1-0472 0-1449 60 1-0472 0-2249-70 1-2217 0-0293 70 1-2217 0-1002-76 38 20 -1-3367 -0-001

78 31 20 -1-3704 0-0002 78 31 20 1-3704 0-0002

The figs. 2, 3, 4 (Plate 8) represent the results of Table II.

In the curve figure (2) the ordinates are the pressures, and the abscissae the arcs

corresponding to 6,

In the curve figure (3) the ordinates are the same plotted to absciss9e=:Il sin 0.

In the curve figure (4) tlie horizontal ordinates are the same as the vertical ordi-

nates in figs. 2 and 2, and the vertical abscissae axe=R cos 0.


These theoretical results will be further discussed in Section IX., where they will

be compared with Mr. Tower's experiments.

29a. 0= '5 is the limit to this Method of Integrating.

In the case considered, in which 6=78"^ SV 20", Table II., shows that the pressure

tow^ards the extreme offside is just becoming negative. With greater values of c this

negative pressure would increase according to the theory.

The possibility of this negative pressure would depend on whether or not the

extreme off edge of the brass was completely drowned in the oil bath, a condition not

generally fulfilled, and even then it is doubtful to what extent the negative pressure

would hold, probably not with certainty below that of the atmosphere.

With an arc of contact anything like that of the case considered it would be neces-

sary, in order to proceed to larger values of c than 5, that the limits between which

the equations have been integrated would have to be changed from

to

2(</>o~|)~<^i. +^1-

This integration has not been attempted, partly because it only applies, in the case

of complete lubrication, when the value of c. > '5 renders approximation very laborious,

but chiefly because it appears almost obvious that the value of c, which renders the

pressure negative at the off extremity of the brass is the largest value of c under

which lubrication can be considered certain.

The journal may run with considerably higher values of c, the continuity of the film

being maintained by the pressure of the atmosphere, which would be most likely to be

the case with high speeds. But although the load which makes c= '5 is not neces-

sarily the limit of carrying power of the journal, it would seem to be the limit of the

safe working load, a conclusion which, as will appear on considering Mr. Tower's

experiments, seems to be in accordance with experience.

This concludes the hydro-mechanical theory of lubrication so far as it has been

carried in this investigation. There remain, however, physical considerations as to

the effect of variations of the speed and load on a and /x which have to be taken into

account before applying the theory.

208 PBOIESSOB 0. EEYNOLDS ON THE THEORY OF LUBRICATION

Section VIIL

—

The Effects of Heat and Elasticity.

30.—-/x and a are only to be inferredfrom the experiments.

The equations of the last section give directly the friction, the intensity of pressure,

and the distance between the cyhndrical surfaces, when the velocity, the radii of

curvature of the journal and the brass, the length of the brass, and the manner of

loading are known {i.e,^ when U, K, a, O^, L, and jut are kuown) ; and, further, if M the

moment of friction is known, the equations afford the means of determining a when /x

is known, or jx when a is known.

The quantities U, B, 6^, and L are of a nature to be easily determined in any experi-

ment or actual case, and M is easily measured in special experiments, but with a and

/x it is different.

By no known means can the difference of radii (a) of the journal and its brass be

determined to one ten-thousandth part of an inch, and this would be necessary in

order to obtain a precise value of a. As a matter of fact even a rough measurement

of a is impossible. To determine a, therefore, it is necessary to know the moment of

friction or the distribution of pressure ; then if the value of /x be known by experi-

ments such as those described for olive oil (Section II.), a can be deduced from the

equations for any particular value of /x, But although the values of /x may have been

determined for all tempei-atures for the particular oil used, and that value chosen

which corresponds with the temperature of the oil bath in the experiment, the ques-

tion still arises whether the oil bath (or wherever the temperature is measured) is at

the same temperature as the oil i|lm. Considering the thinmass of the film and the

rapid conduction of heat by metal surfaces, it seemed at first sight reasonable to

assume that there would be no grea-t difference, but when on applying the equations

to determine the value of a for one of the journals and brasses used in Mr. Tov/er's

experiments, it was found that the different experiments did not give the same values

for a, and that the calculated values of a increased much faster with the velocity when

the load was constant than with the load when the velocity was constant, it seemed

probable that the temperature of the oil film must have varied in a manner unper-

ceived, increasing with the veloqity and diminishing the viscosity, which would

account for an apparent increase of a.

That a should increase with the load was to be expected, considering that the

material of both journal and brass are ela^stic, and that the loads range up to as much

as 600 lbs. per square inch, but there does not appear any reason why a should

increase with the velocity unless there is an increase of temperature in the metal. If

this occurs, the apparent increase of a would be part real and part due to the unappre-

ciated diminution of /x owing to the rise of temperature.

Until some law of this variation of temperature and of the variation of a with the

load is founds the results obtained from the equations, with values of /x correspond]n^


to some measured temperature, such as that of the oil bath ot a point in the brass, can

only be considered as approximatisly appHcable to actual results. Even so, hoAvever,

the degree of approximation is not very wide as long as the conditions are such that

the journal *'runs cool.''

But, treated so, the equations fail to show in a satisfacttfry way what is one of the

most important matters connected with lubrication—the ciricumstances which limit the

load which a journal will carry. F.or, although it may be assumed that the limit is

reached when ca, the shortest distance betweeii the surfaces,.becomes zero or less than

a certain value, yet, according to the eqfiiations, assuming a and /x to be constant, the

value of c increases directly as U if the load be constant ; so that the limiting load

should increase with U. But this is not the case for it seems from experiments that

at a certain value of U the limiting load is a ma:ximum if it does not diminish for a

further increase of U.

Although, therefore, the close agreement of the calculated distribution of the

pressure over the bearing with that observed and the app'roximate agreement of the

calculated values of the friction for different speeds and loads, such as result when fi

and a are considered constant, seem to afford sufficient Verification of the theory, and

hence a sufficient insight into the general action of lubrication, without entering into

the difficult and somewhat conjectural subject of the effects of heat and elasticity,

yet the possibility of obtaining definite evidence as to the- circumstances which deter-

mine the limits to lubrication, which, not having been experimentally discovered, are a

great desideratum in practice, seemed to render it worth while making an attempt to

find the laws connecting the velocity and load with a, and /x.

As neither the temperature of the oil film or the interval between the surfaces can

be measured, the only plan is to infer the law of the variations of these quantities

from such complete series of experiments as Mr. Tower's. In attempting this, a

probable formula with arbitrary constants is first assumed or deduced from theoretical

considerations, and then these constants are determined from the experiment and the

general agreement tested. In order to determine the actual circumstances on which

the constants depend, it is important to obtain the formula from theoretical considera-

tions. This has therefore been done, although these considerations would not be

sufficient to establish tlie formulae without a close agreenient with the experiments.

31. The Effect of the Load and Velocity to alter the Value of the Difference of Radii

of the Brass and Journal, i.e., of a.

The effect of the load is owing to the elasticity of the materials, hence it is probable

that the effect will be proportional to the load L. To express this put

—

a^a^+mL (109)

MDCCCLXXXVI. 2 E

210 PEOPESSOU 0. REYJS^OLDS ON THE THEOEY OF LUBRIOATIOJST

The eifect of the temperature on a is owing to the different coefficients of expansion

of brass and iron. Thus :

—

where B' and S' are the respective coefficients of expansion of the bearing and journal.

These for brass and iron are :

—

B'= -0000111

S'=-0000061,

therefore putting T--To=Tm (the mean rise of temperature due to friction)

aT=aofl+ '000005—Tm

Putting=ao(l+ETm) (110)

E for -000005a^

If, as seems general, a^ is about '0005 inch, then with a 4-inch journal

E=-02 about (Ill)

which is sufficiently large to be important.

32. The Effect of Speed on the Temperature.

Putting

—

Tq the temperature of the surroundings and bath..

Tj the mean temperature of the oil as it is carried out of the film.

1l^ the mean rise in temperature of the film due to friction.

Q the volume of the oil carried through per second.

D the density.

S the specific heat.

H the heat generated.

Hj^ the heat lost by conduction.

C a coefficient of conduction.

taking the inch, lb. and degree Fahr. as units, and 12 J as mechanical equivalent of

heat,

2

H= -

—

-—

-

12J

H~H"1 -^o~ DSQ

•


Putting

H~Hi=gT,,Q . (112)

where g' is a constant depending on the relative values of T

—

Tq and T„i, also on how

far the metal of the journal assists the oil in carrying out heat.

On substituting the values of H^, Hg, Q> it appears

^^^SJDS^îU -p' (112a)

B=U3C'

There does not appear to be any reason to assume any of the quantities in the

denominator to be functions of the temperature except h^. By equation (42)

Equation (112a) may thus have the form

-*- m— ^^ • • • • (Îloj

A(l + ET,) +gWuere

A=3JDSgao(l+mZ){l+csin(<î--<ô)} (114)

This shows that A is a function of the load, increasing as the first power and

diminishing as the second power, but the experiment show the effects of these terms

are small, and A is constant except for extreme loads.

33. The Formulcefor Temperature and Friction^ and the Interpretation of the Constants.

From equation (8), Section II,

C=0221 (for olive oil) J

^


a=(ao+mZ){l+E(T--To)} , , , (116)

or approximately since E(Ti— Tq) is small

a=:(ao+m?)e^^'^"*'^'^^ . . . . . . . - . (117)

2 E 2

212 PROFESSOR 0. REYNOLDS OlST THE THEORY OF LUBRICATION

Whence substituting in the equation which results from (51) c being small

/=^U (118)

Putting T^ for any particular temperature /x^ a^ corresponding values

yL^_J^^^XJ6-{C+ E}(T-T.) (119)

From (113)

/=A{T,,+ET^4+|t,, (120)

These equations (119) and (120) are independent, and therefore furnish a check

upon each other when the constants are known.

Thus substituting the experimental vahies of U and /in (120) a series of values of

T^ are obtained, which when substituted in (119) should give the same value of/. In these equations the meaning of the constants is as follows :-—

C is the rate at which viscosity increases with temperature.

E is the rate at which a increases with temperature, owing to the different ex-

pansion of brass and iron.

AU expresses generally the mechanical equivalent of the heat which is carried

out of the oil film by the motion of the oil and journal for each degree rise of

temperature in the film.

B expresses the mechanical equivalent of the heat conducted away through the

brass and journal.

The respective importance of these two coefificients is easily apparent. When the

velocities are low but little heat will be carried out, and hence the temperature of the

journal depends solely on the value of B. But when the velocities are high, Bbecomes insignificant compared withAU, and it is A alone which keeps the journal cool.

The value of A may to some extent be inferred from the quantities which enter into

it as in equation (114). Thus in the case of Mr. Tower's experiment, since

E=2 (9zz:l-37 a^-00075 J=772D= 0-033 S= 0*31 (for olive oil)

A=*0063g (121)

It is very difficult to form an estimate of g, but it would seem probable that it has a

value not far from 2 ; and, as will be subsequently shown, in the case of Mr. Tower'sexperiments, j is about 3*5 or

A= 0-0223 (122)

AND ITS APPLTCATIO:^ TO MR. B. TOWER'S EXPERIMENTS. 213

As B expresses the rate at which the heat generated in the oil fihn is carried away by

conduction through the oil and the surrounding metal, any estimate of its value is very

difficult. If we could measure the temperature at the surfaces of the metal, B might

be made to depend only on the thickness and conductivity of the oil fihn. But before

heat can escape from the journal or bearing it must pass along intricate metal channels

formed by the journal or shaft and its supports ; and, on consideration, it appears that

in ordinary cases the resistance of such channels would be much greater than the

resistance of the oil film itself. For example, in the case of a railway axle, the heat

generated must escape either along the journal to the nearest wheel or through the

brass and the cast-iron axle box to the outside surface, so that either way it must

traverse at least three or four inches of iron. This is about the best arranged class of

journals for cooling. In most other cases" heat has much further to go before it can

escape. However, in every case B will depend on the surrounding conditions and can

only be determined by experiment. From the experiments, to be considered in the

next section, it appears that

B= l (about) (123)

But it is to be noticed that Mr. To^ver has introduced a somewhat abnormal con-

dition by heating the oil bath above the surrounding temperatui^e. For in this way,

letting alone the heat generated by friction, there must have been a continual flow of

heat from the bath along the journal to the machinery ; and, considering the compara-

tively limited surface of the journal in contact with the hot oil and the large area of

section of the journal, it appears unlikely that the temperature of the journal was

raised by the bath to anything like the full temperature of the latter, a conclusion

which is borne out by Mr. Towner's experiments with different temperatures in the

bath (Table XII. > Art. 34), which shows that the temperature of the bath produced a

much smaller effect on the friction than would have followed from the known viscosity

of oil had the temperature of the oil film corresponded with the temperature of the

bath.

Thus the temperature of the film independent of friction is not the temperature of

the bath or surrounding objects, and as it is unknown until determined from the

experiments, it will be designated as

T

and the suffix x used to designate the particular value for T=T;^.of all those quantities

which depend on the temperature as

If T be the mean temperature of the film,

where T,;^ is the rise of temperature due to the film.

214 PROFESSOR 0. REYNOLDS ON THE THEORY OF LUBRICATION"

33a. The maximum Load the Journal will carry at any Sêed,

It Las already been pointed out that the carrying power of the journal is at its

greatest when c is between '5 and '6. If, therefore, taking the load constant, c passes

through a minimum value as the velocity increases with a constant load, then the load

which brings c to a constant value will be a maximum for some particular velocity,

and if the particular value of c be that at which the carrying power is greatest, the

carrying power will be greatest at that particular speed.

The question whether^ according to the theory, journals have a maximum carrying

power at any particular speed turns on whether

— (c being constant)

is zero for any value of U.

This admits of an answer if the values for /x, a, T^, and equations (119) and (120)

\j fhold, for when c is constant ^^nFr? tFtt? ^^^ ^ ^^^ constant, whence differentiating and

substituting it appears that when c is constant

B /,_ ,.^ _BA+^4-fAE^(3E + C)^)TÂE^T^

^ ^^^ — .... (123b)

^^ Â +^+ |(3E + C)A-f(E + C)?-!.T+AE(E + C)T2

where U is to be taken positive, and T increases as U increases. This shows that

7—, for a constant value of c, changes sign for some value of U if T continues tod\j

00increase with U.

Hence, according to the theory, the values of L, which make c constant as Uincreases, approach a maximum value as U increases, and since this value, when c is

about '5, represents the carrying power of the journal, this approaches a maximum as

IT increases.

Section IX.—Application of the Equations to Mr. Tower's Experiments.

34. References to Mr. Tower's Reports,

From the experiments described in Mr. Tower's Reports I. and II., in the Minutes

of the Institution of Mechanical Engineers, 1884, the journal had a diameter of

4 inches, and the chord of the arc covered by the brass was 3" 92 inches, the length of

tjie brass being 6 inches.

The loads on the brass in lbs., divided by 24, are called the nominal load per

square inch.

AND ITS APPLIOATIOK TO MR. B. TOWER'S EXPERIMENTS. 215

The moments of friction in inches and lbs., divided by 24 X R, are called the nominal

friction per square inch.

These nominal loads, and nominal frictions, with the number of revolutions from

100 to 450, at which they were taken, are arranged in tables for each kind of oil used,

and also for the same oil at different temperatures of the bath.

All the tables relative to the oil bath in the first report refer to the same brass and

journal. And with this brass, to be here called No. 1, no definite measurements of

the actual pressure were made.

The second report contains the account of the pressure measurements, but it is to

be noticed that these were made with a new brass, here called No. 2, and that the

only friction measurements recorded with this brass are three made at velocities five

times less than the smallest velocity used in the case of brass No. 1.

It thus happens that while by the appHcation of the foregoing theory to the friction

experiments on brass No. 1 a value is obtained for a, the difference in radii of brass

No. 1 and the journal ; and from the pressure experiments on brass No. 2 a value is

obtained for a in the case of brass No. 2 ; since these are different brasses there is no

means of checking the one estimate against the other.

The following tables extracted from Mr. Tower's reports are those to which

l*eference has chiefly to be made.

The first of these extracts is a portion of Table I. in Mr. Tower's first report ; this

related to olive oil, but corresponds very closely with the results for lard oil also,

although not quite so close for mineral oil.

The second extract is Table IX. in the first report ; this shows the effect of

temperature on the friction of the journal with lard oil.

The third extract is from the second report, being Table XII., representing the oil

pressu.re at different parts of the bearing as measured with brass No. 2.

From Table I. (Mr. Tower's 1st Report, Brass No. 1.) Bath of Olive Oil. Tempera-

ture, 90 deg. Fahr., 4-in. journal, 6-in. long. Chord of arc of contact=3 '9 2 in.

nal

load

per

square

h.

Total

d

divided

4x6.

Nominal friction per square inch of bearing.

100 rev. 150 rev. 200 rev. 250 rev. 300 rev. 350 rev. 400 rev. 450 rev.'5 ta o ce ^ 105 ft. 157 ft. 209 ft. 262 ft. 314 ft. 360 ft. 419 ft. 47 J ft.O 1—'

""I "—^ r"^

per mjn. per min. per mm. per mm. per mm. per mm. per mm. per min.

520 • ••416 •520 •624 •675 •728 •779 •883

468 • 1f•514 •607 •654 •701 •794 '841 •935

415 • 4•498 •580 •622 •705 •787 '870 •995

363 ••472 •580 •616 •689 '725 •798 •907

310 • ••464 •526 •588- •650 •680 •742 •835

258 •361 •438 •515 •592 •644 •669 •747 •798

205 •368 •43 •512 •572 •613^

^675 •736 •818

153 •351 •458 •535 •611 •672 •718 •764 •871

100 •36 •45

i

•555 •63 •69

1

•77 •82 •89

216 PROFESSOR 0. REYNOLDS ON THE THEORY OF LUBRICATIOK

From Mr. Tower's 1st Keport, Brass No, 1.

Table IX. Bath of Lard Oil. Variation of Friction with Temperature. Nominal

Load 100 lbs. per square inch.

Temperature.Jbahr.

Nominal friction per square inch of bearing.

100 rev.

per niin.

1.50 rev.

per min.200 rev.

per min.250 rev.

per min.300 rev.

per min.

3f'0 rev.

per min.

400 rev.

per min.450 rev.

per min.

12011010090807060

•24

•26

•29

•34

•4

•48

•59

•29

•32

•37

•43

'52

•65

•84

'35

•39

•45

•52

•63

•8

1-03

•40

•44

•51

•60

•73

•92

1-19

•44

•50

•58

'69

•83

1^03

1-30

•47

•55

•65

•77

•93

M51-4

•51

•59

'71

•85

1-02

1-24

1'48

•54

•64

•77

•93

1-12

1-33

1-56

From Mr. Tower'b Second Report, Brass No. % Heavy Mineral Oil. Nominal Load

333 lbs. per square inch. Number of revolutions 150 per minute. Temperature, 90^^

Table XIL Oil pressure at differents points of a bearing.

Longitudinal planes

Pressure per square inch

,

Transverse plane, middle

.

Transverse plane, No. 1 .

Transverse plane. No. 2 ,

on centre off

lb. lb. lb.

870 625 500

355 615 485

310 565 430

The points at which the pressure was measured were at the intersections of six

planes, three parallel vertical longitudinal planes parallel to the axis of the journal,

one through the axis, and the other two, one on the on and the other on the o^side,

both of them at a distance of '975 inch from that through the axis, three transverse

planes, one in the middle of the journal, the other two respectively at a distance of

one and two inches on the same side of that through the middle.

In referring to these experimental results in the subsequent articles.

The nominal load per square inch is expressed by \I

;

The number of revolutions per minute by N^

;

The nominal friction by f[

;

The effect of the fall of pressure at the ends of the journal on the mean pressure

1is expressed by-, thus—


n is a coefficient depending on the way in whicli the journal fits the shaft.

L'

/'

N'=

s

_L

4s

wM4R

Ux602'n-ll

~\

> « s • • » 5 • (124)

1-21-y

35. The Effect of Necking the JournaL

The expression (124) fory assumes that the journal was not necked into the shaft.

From Mr. Tower's reports it does not appear whether or not the brass was fitted into

a neck on the shaft ; but since there is no mention of such necking, the theory is

applied on the supposition that there was not.

If there w^ere, the friction at the ends of the brass would increase the moment of

friction. Put b for the depth of the neck and a' for the thickness of the oil film at

the ends, then the moment of resistance of these ends would be—

a'^ 30n+r^'K

Hence if M be the moment of friction of the cyKndrical portion of the journal only

-/'-€f<«+r-f • • 4 a t • (125)

And from equation (97)

—

r-

J(126)

For example, a 5-inch shaft necked down to a 44nch journal would give 6=:'5 inch.

Whence, assuming

a'~-0005

and1^

a

a'

^— &i

.... (1 ^ i

\

the relative friction of the ends to that of the journal would be 11*36 to 31*00, or

MDCCCLXXXYI. 2 F

218 PROFESSOR O. REYNOLDS ON THE THEORY OF LUBRIOATIOIS"

28 per cent- of the friction ; and the values of a^ calculated on the assumption of no

necking, would have to be increased in the ratio n=l*33.

Even if there is no necking the value of a will probably not be the same all along

the journal, in which case the values of a^ aud a in K^ and K^ will be means, and then

the square of the mean will be less than the mean of the squares, so that n will probably

have a value greater than unity^, although there raay be no necking of the shaft.

36. A first Approximation to the Difference in the Radii of the Journal and

Brass No, 1.

The recorded temperature in Mr. Tower's Table I, is 90°F^hr. Accepting this, and

taking the value of /x, equation (8), Section II.,

^9q::=:1G~^X6'£1 s • • ^ fl 9 . . (128)

By equation (97)-—

Since 11=3 and IJ:

fifjb —^Ma 2'75R2U

« » » ^ a »

a

•72/'

U

. , . (1 ^t/

)

= 8-46z3 9 « . (130)

Whence substituting from equation (128) for /x90

2= 10-6 X 1-975

n J9 5 »

"

« (131)

and from the tabular Nos, for L'=100,

N=100, /=-36

a

nIC^XS-S (inch)

N=480, /=-89> Add

a

n-=IQ-^X10 (inch)

. .. (132)

These are the extreme cases ; for intermediate velocities intermediate values of -

are found.

» a

71


In order to be sure that these are the values of -, which result from the application

of the equations, it is necessary (since the approximate equations only have been

used) to see that the squares of c may be neglected.

Substituting from equation (124) in (96)

/Rwhich for L'=:100, N= 100, gives

and for L'=100, ^= 450,

* » • 4> « « . . . (133)

So that the approximations hold, and, as already stated in Art. 30, this consider-

able increase in the value of a with the load constant suggests that the temperature

of the film was not really 90°. And as this point has been considered in the last

section the equations of that section may be at once used to determine the law of

this temperature, after which the values of - may be determined with precision.

37. The Rise in Tewperattire of the Film owing to Friction,

In order to determine the values of E, A, and B in equations (119) and (120), by

substituting in these equations corresponding values of N and /' for L^=100, the

tabular values of/' were somewhat rectified by plotting and drawing the curve N, /'

.

These corrected values are in the second row, Table III.

From these values, and the corresponding values of N, it was then found by trial

that the equations (119) and (120) respectively

c=*0221

are approximately satisfied for values of T—T^=T„, if

A= '022308

E=-0222

B=-95914

^ F ^

o««;>s«t

r" « « * t « < \ \ tJ *jj

220 PROFESSOR O. REYNOLDS ON THE THEORY OF LUBRICATION

Which seemed to agree very well with the reasoning Section VIII. With these

values of the constants the values of T— T^, were then calculated from equation (120),

and are given in the third row in Table III. These temperatures were then substi-

tuted in equation (119), and the corresponding values oif calculated, these are given

in the fourth row. Table III.

Table III.—Rise of Temperature in the Film of Oil caused by Friction, calculated

by Equation (120) from Experiments with a Nominal Load 100 lbs. (see

Table L, Tower, Art. 34).

Kominal friction per square inch, as calculated by equation (119) from the rise of temperature.

_ <D O

a |ji

O

/T-T„Fahr.

/i

Revolutions per minute ....Nominal friction

[Table I., Tower

per square inch < Corrected to a

for olive oil [ curve . . .

Rise of temperature by equation

(120) .

Nominal friction per square inchcalculated by equation (119)assuming c small

100 150 200 250 300 350 400 450•36 •45 •54 •63 •69 •77 •82 •89

•33 •55 •55 •63 •705 •768 •83 •89

3-45 5-83 8-13 10-02 11-77 13-26 14^48 15-37

•386 •453 •546 •628 •697 •76 •823 •89

The agreement between these calculated values off and the exj)erimental values

is very close ; and it may be noticed that a very small variation in any of A, B and Emakes a comparatively large difference in some one or other of the calculated values

of y^, or^ in other words, these are the only positive values of these quantities which

satisfy the equations.

The only dijfference between the experimental and calculated values of f which is

not explainable as experimental error, is for the lowest speed at which the

experimental value of /' is 6*7 per cent, too large. This is important as it is in

accordance with what might be the result of neglecting c^, since at that speed

c is becoming too large to be neglected, and taking c^ into account the calculated

value oif agrees very closely with the experimental.

38. The Actual Temperature of the Film.

Having found the approximate values of T^, the rise of temperatui'e, owing to

friction, it remains to find T^, the temperature of the film, the rise due to friction,

so that

T^H-T^=: temperature of film.

This is found from Mr. Tower's Table XI. (see Art. 34).

Putting

Tb for temperature of the bath

Tq for temperature of surrounding objects

and assumingr^^,-}~T?%^^^^( M3— Io)"t" -l?^"!" 1-0 <...'.'• (136)

AKD ITS APPIilOATIO:^" TO MR, B. TOWER'S EXPERIMENTS.


•whence

^ 3*46 6^0 4-mL

log/=-0443{Z(TB^To)+T.}log«+log(3^^^). . . (137)

In Table IX. (Tower) the values off are given for the same vahies of W and L'

corresponding to different values of Tb.

Substituting corresponding values of^ and Tb in equation (137), and subtracting

the resulting equations, we have an equation in which the only unknown quantities

are Z and the differences of T^.

The values of/' being known, the values of T^ are obtained from equations (120)^

(134), and (125), and substituting these, the equations resulting from (137) give the

vahies of Z. Thus from Table IX. (TowEn)

From (137)

L'=100

u=ioo

Tb=60 /—59 ^m y

Tb=70 /-•48 T,,"4-8

-4-8_ log -59

10 ""•0443x

-log -48

10 X log e

Therefore/La OOe e * . » * • » • • . (I O I

From this value of Z the values of /' corresponding to those in Towee's Table IX.

have been calculated and agree well with the experimental values.

The smallest temperature of the oil bath recorded in Tower's Table IX. Is 60^ Fahr.,

therefore it is assumed that this was the normal temperature, whence

T^=-35(Tb-60)+ 60 ,.,,..., (139)

Hence it is concluded that the actual temperature of the oil film in all the experi-

ments with the bath, at a temperature of 90^ Fahr., is given by

T=:70'5+T,« ,,,,.,..,. (140)

By the formula for \l since T^=70*5

;x,-*00004737e^'00^i^^«'^

= ^000009974 .....,.«.. ^ (ill)

:^-*00001 (approximately)

222 PEOFESSOR 0. REYNOLDS ON THE THEORY OF LUBRICATION

and since^ by equation (I34)j when L'=100

= •000741372 (142)

= *00074w (approximately).

This is the value of a^ with a load of 100 lbs. per square inch.

39. The Variation of a ivith the Load.

All Mr. TowEE^s experiments^ when the loads are moderate and the velocities high/

show a diminution of resistance with an increased load.

Since c increases with the load and the friction increases as c increases, a and fi

being constant^ the diminution of friction with increased loads shows either that the

load increases the temperature of the film and so diminishes the viscosity, or increases

the radius of curvature of the brass as compared with that of the journal, i.e.,

increases a.

These effects have been investigated by substituting the experimental values of f^

and 1a\ obtained with the same velocity in equations (119) and (120).

In this way, from equation (119) the value of m is determined, where, from

equations (117) and (135),

a^={aQ+mV)ne^-^^^'^-"'^'K . (143)

And the equation (120) gives the effects of the load on the value of the constant A.

After trial, however, it appears that the effects of the load upon the constant A are

small so long as the loads are moderate, and that the diminution of the resistance

with the increased load is explained by the value obtained for m from equation (119).

From this equation, taking L^^ /^i, L^g /^^, simultaneous values of L' and f\ and

assuming T^ independent of the load

^+^^4"^/^^ .......... (144)

which gives the value of m.

The slight irregularities in the experiments affect the values of m thus found to a

considerable extent, and a mean has been taken, which is

m= -002^0 ....,.,.., (145)

Putting T^=70*5, Tq— 60, ce^=*00074, when L'=:100, from equation (143)

ao4-mL'=^ '0005861^,

Therefore

^0=^0004885^ (146)

AED ITS APPLICATION TO MR. B. TOWER'S EXPERIMENTS. 223

The value for a thus obtained is therefore

a= •0004885^(1+ -002 L')e''"''<^»+'^»--''°' . . . . (147)

and for tlie experiments with the bath at 90° Fahr.

a=-0004885^i(l+ -002 y^^'^'^'^^^u^^^'^)

or

a=-0006161r?.(l+ -002L>' 0222Tm.

t •

From equation (148) and the values of T^, Table Til,, the values of

n

for Mr. Tower's experiments have been calculated.

Putting

and, substituting in equations (47) from (148) and (149),

K,:

-K„=

66-08 Ne-o'^'^sTm

(1 + -002 L')V0034N6-'o«8T„

2'

(1 + -002 ll)n >

K̂?=: -00005139(1+ -002 L>0222T

(148)

(149)

(150)

for. the circumstances of Mr. Tower's experiments, to, which the equations of

Sections VI. and VII. then become applicable.

40. Application of the Equations to the Circumstances 0/ Mr. Tower's Experiments

on Brass No. 1, given in Table I., Art. 34, to determine c, j>.q, 4>i>/' and p—p^.

The circumstances are, the unit of length being the inch,

11=2 (inches)

6>=78° 31' 20"

1

«(,,

»•0004885, already deduces, equation (146)

L=484 L'

UT

equations (124)y. (151)

0'

T:V

T

•2094

60, assumed

70*5, equation (140)

tabular values, Table III., the increase with load being neglected

224 PEOFESSOR O. REYNOLDS ON THE THEORY OE LUBRIOATION

For a first Approximation as long as c is small.

Equations (94), (89), (91), and (95) are used to determine o, <^(^, 4*>f\ f^^' ^^^ experi-

ments in Table I. (Tower), these being made with brass No, 1,

Putting as in equation (124)

r-

equation (94) gives

and by equation (150)

A'=-l-Z7'K,n,

]S[e-044ST

//= -004658^^^:^21' ('^2)

From equation (152) the vahies of/^ to a first approximation have been calculated,

using the values of T^ given m Table III. These are given as/^^ in Table lY*

Table IV.—Olive Oil, Brass No. 1.

LeHgth of the journalChord of the arc of contact of the brass

Eadiusof the journaJ. ...Temperature of the oil bath

,f surrounding objects

Difference in radii of brass and journal at 60°,

Effect of uecMug or variations in radius to increaBe friction . . . ,

6 inches.

3'92 inches.

2 inches.90° Fahr.60° Fahr. (asaumed\0-0006 inch (deduced).1-25.

nominal load in lbs. per square inch, being the total load divided by 24.

number of revolutions per minute.nominal friction in lbs. per square inch from Table I. in Mr. Toweb's first Report (see Art. 34).nominal friction calculated by complete approximation for c= 6 (see Art, 40, equation (169j).nominal friction calculated to a second approximation, equation (154).nominal friction calculated to a first approximation, equation (152).rktio of the distance between the centres of the brass and journal to the difference in the radii, equation (153).difference in the radii of the brass and journal (see equation (157) ).

angular distance from the middle of the arc of contact of the point of maximum pressure, equation (91).

angular distance from the middle of the arc of contact of the point of nearest approach (see equation (89) ).

Tot the rise in temperature of the film of oil owing to the friction, equation (120).

V the

N the/' the

(//) the

// the

/,' the

c the

a the

01 the

00-2 the

N. 160 150 SOO 250 300 350 400 450

Tm Fahr. 3-45 5'8a 8'13 10'02 ir77 13-26 14^46 15^37

r/' « • '498 •680 •622 •705 •787 •870 •995

(//) ft • « * (-57) (•65) • * • • * *, //no8/i' • • '300 •360 •414 •460 •504 *544 •589

L'=415 ic * • 67 •578 •520 •487 •457 •436 •413a '00154 '00162 •0017 •00178 •00182 •00187 •00191 •00194

1 01 A • • •-7° 0' (}" — 7° 0' Q" * » '• • -• • •

• • * * - 42° 0' 0" -42°0' 0'^« 9 • • • • «

r/' • • •472 •580 •616 •689 •725 •798 •907

(//) ""' % (•498) «

•

«

•

• * • //•920; // •233 •314 •378 •434 •482 •526 •573 •616

L' = 863 ^ ^ • • •520 •46-2 •40S •380 •357 •340 •312« • •00151 •00161 •00168 •00171 •00177 •00181 •00183

: -Pi •-7° O'O'' • • * • .

.

* *

'

• * • •

.^«-2 » -42° 0' 0"

• • « * » e » « • • • •

A^B ITS APPLICATION TO MR. B. TOWER'S EXPERIMENTS. 225

Table IV.—Olive Oil, Brass No, 1—continued.

N. ICO 150 200 250 300 350 400 450

Tm Fahr, 3'45 5-83 8-13 10-02 11-77 13-26 14^46 1537

'/'. • •464 •526 •588 •650 •684 •742 •835

(/lO ('370) • • • • • • • • //•765 •805 •845

y.' •249 •336 •404 •464 '517 •582 •610 •660

I/ == 310 -\

c •510 •392 •337 •305 •284 •266 •254 •234

a ^00138 '00144 '00151 •00158 •00161 '00166 •00172 •00172

^1 - 7° 0' 0''• • • • • « • • « • • • • •

1

-42° 0' 0''• • • t • • » • • • • • • •

r/' •361 •436 •514 •592 •644 •669 •747 •789

// • • • • • ••62 •670 •712 •760 •810

/i' •266 •358 '431 '495 •550 '600 •650 •707

L == 253 <c •377 '287 •242 •224 "200 •195 •180 •171

a •00128 '00135 •00141 •00148 •00151 '00156 •00159 •00161

<Pi • • • • • • • • »

•

• • • • • •

A. "^• • • • • • • • • • * • • • •

ff '368 •430 '512 •572 '613 ^675 •786 •818

1 // •380 '457 •530 •595 ^Q5 •701 •755 •810

/>' •285 •385 '464 •£34 '592 '646 '700 •755

y-1 /• •255 •195 •165 •162 •141 •132 '126 •118

= 2sU0 •<

a •00119 •00126 •00131 •00138 •00240 •00145 •00148 •00135

^i • • • • • • • • • • • •^1° 13' 0" -1° 5' 0"

L*°~2 * • • • • • • • • t • •

-60°0' 0" - 62° 0' 0"

r/' •351 '458 •535 •611 •672 '718 •764 •871

// '352 •458 •530 '601 '665 •717 '778 •840

/i' •307 •414 '498 •574 •6^8 •695 •753 •813

v==168 -c •165 •126 •107 •098 '089 '086 •082 •075

a •00111 '00116 •00122 •00128 •00130 •00134 •00137 •00139

^1 • •-1° 13' 0^' -1° 0' 0" -0° 57' 0" -0° 50' 0" -0° 44' 0" -0° 38' 0" -0° 31' 0"

,^»-| • • -ero' 0'' -65° 0' 0" -2T°20'G" -69°0' 0" -69°40'0" -70°20'0" -72''0' 0"

r/' '360 '450 '550 •630 •690 '770 •820 •890

// •352 '465 •555 •637 •T08 '770 '8 a •897

// '336 •463 •546 •628 •697 •760 •823 •890

v== 100 .c •090 •0691 '0600 '0541 •0492 •0471 •0460 •0420

a •00101 '00106 •00112 •00116 •00120 '003 23 •00125 •00127

^1-0° 43' 0'' -0° 26'

0'' -0° 17'0" -0° lO'O" -0° 4' 0" -0° 2' 30" -0° 0' 0" + 0° 5' 0"

-68°30'0'' - 73° 20' 0" -75°20'0" -76° 30/0" -7r°20'0" -.77°40'0" -0° 78' 0" -78°40'0"

As compared with the experimental values f^ given in the liable IV., it is seen

that the agreement holds as long as c is less than '06, after whicli, as c increases, the

values oi f^ become too small, or while the values oi fi continue to diminish as the

load and a increase, the experimental values of /' after diminishing till c is about

•1 or '15 begin to increase again. In order to see how far this law of variation was

explained by the theory, it was necessary to find f^ the values of /' to a second

approximation, and before this to obtain the values of c.

Putting, as in equation (124),

L=4-84L',

MDCCCLXXXVI. 2 G

226 PROFESSOR 0. RErNOLDS OIsT THE THEORY OF LUBRICATION

equation (94) gives

and by equation (150)

c=-03116(l+ -1002LO^'^|^V'^^^^ , (153)

Equation (153) gives the values of

n^'

QTo obtain the value of n from the experiments, these values of —̂ are substituted in

the equation for/*", retaining the squares of c, which obtained from equations (89) and

(85), is

//=//(! +5^'). • • (154)

whence, substituting the values of -^obtained from (153) we have

//=/i'+5(f,)V/,'.

Therefore, choosing any experimental values of /', and subtracting the corresponding

value of fi in Table IV., n is given by

—

it - -'" 7 r~o • • • « » o • • • • \ 1. tJ O e

In this experiment irregularities become important, and it has been necessary to

calculate several values of n in this way and take the mean, which is

It has been shown (Art. 35) that necking might account for a value of n as great as

1*33, while if there were no neckings would still have a value in consequence of

variations of a along the journal.

Substituting this value oi n in equations (148 and 153),

/\^-0222T»t "^

a=:-00077(l+ -002 L>

c=;-0487(l+ -002 L') ^e"^"''"!" . e . • o o o \-L V I /

from which equation the values of a and e have been calculated for Table IV. for all

values of L' less than 415 lbs. These are all Mr. Tower's experiments with olive oil,

except those of which Mr. Towee, has expressed himself doubtful as to the results.

AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 227

The values of c, as given by equation (94) are only a first approximation and are too

large, but the error is not large, even when c='5 only amounting to 8 per cent., as is

shown by comparing equation 104 with 95.

With these values of c in the equation (154) the values of /g' have been calculated

for all values of c up to '250. At c= *12 these values o{ f^ are about 5 per cent,

larger than the experimental values, but they have been carried to c='25 in order to

show that the calculated friction follows in its variations the idiosyncracies of the

experimental frictions, falling with the load to a certain minimum, and then rising

again.

These values of f^^ carry the comparison of the frictions deduced from the theory

up to loads of 205 lbs. for all velocities, and up to 363 lbs. for the highest velocity.

To carry the approximation further use has been made of .the more complete inte-

grations of the equations for the case of

c='5.

These are given by equations (104) and (105).

As already stated, comparing (104) with (94) it appears that when c='5 the

approximate values of c in the Table IV. are about 8 per cent, too large ; that is to

say, a value c=*540 in the table would show that the actual value was c=:*5.

Comparing equation (95), from which the values f^ have been calculated, with

equation (105), it appears that when c==*5 the values o{f by (105) are given by

"^ 1-37 -^1

This is not, however, quite satisfactory, as that portion of the friction which is due

to necking does not increase with the load. This portion in f^ is

So that for c=*5

Ji

.,_ 2'3773 + l-370i-l)^ , .

and since 7i==l'25, this gives for c='5

/'=l-585/l'

If, therefore, any of the approximate values of c were exactly '640, the complete

value ofy would be 1*585 times the value of fj. This does not happen, the nearest

approximate values of c being '578, '520, *520, *510, Multiplying the corresponding

values of /^' by 1*585, the results are as follows :—

Z G ^

228 PROFESSOR O. REYNOLDS ON THE THEORY OF LUBRICATION"

Tabular. Tabular.1-605//.

Experimental.DiflPerence.

•578 •36 •57 •58 •01

•520 •414 •656 •65 005•520 •314 •498 •472 •026

•510 •249 •394

It tlias appears that the approximation is very close, the calculated values for the

first, in which c is greater than '540, being too small, and for the rest, in which c was

smaller than '540, too large, which is exactly what was to be expected.

These corrected values of f{ have been introduced iu Table IV. in brackets. As

they occur with different loads and different velocities, they afford a very severe test

of the correctness of the conclusions arrived at as to the variations of A and T with

the load and temperature, also as to the condition expressed by n. Had the values of

c and f been completely calculated as for the case of c='5, there would have been

olose agreement for all the calculated and experimental values oif

,

This close agreement strongly implies, what was hardly to be expected, namely, that

the surfaces, in altering their form under increasing loads, preserve their circular

shape so exactly that the thickness of the oil film is everywhere approximately

a(14-<^ sin (^—^)).

A still more severe test of this is, however, furnished by the pressure experiments

with brass No. 2 in Mr. Tower's second report.

41. The Velocity of Maximum Carrying Power,

The limits to the carrying powers are not very clearly brought out in these recorded

experiments of Mr. ToweRj as indeed it was impossible they should be, as each time

the limit is reached the brass and journal require refitting. But it appears from

Table I. and all the similar tables with the oil bath in Mr. Tower^s reports, that the

limit was not reached in any case in which the load and velocity were such as to make

c less than '5. In many cases they were such as to make c considerably greater than

this, but in such cases there seems to have been occasional seizing. There seems,

however, to have been one exception to this case, in which the journal was run at

20 revolutions per minute with a nominal load of 443 lbs. per square inch with brass

No. 2 without seizing, in which case c, as determined either by the friction or load,

becomes nearly *9.

It does not appear that any case is mentioned of seizing having occurred at high

speeds, so that the experiments show no evidence of a maximum carrying power at a

particular velocity.


This is so far in accordance with the conclusions of Art. (33a), for substituting the

values of A B C E, as determined Art. (37), it appears by equation (123b) that the maxi-

mum would not be reached until T^, the rise of temperature due to friction, reached

72° Fahr., which, seeing that at a velocity of 450 revolutions T^ is less than 17°, implies

that the maximum carrying power would not be reached until the speed was 1500 or

dlj2000 revolutions ; notwithstanding -z~ c constant is very small at 450 revolutions.

This is with the rise of temperature due to legitimate friction with perfect lubrica-

tion. But if, owing to inequalities of the surfaces, there is excessive friction without

corresponding carrying power, ^.e., if n, the effect of necking, is a.s large as 3 or 4,

which it is with new brasses, then the maximum carrying power might be reached at

comparatively small velocities ; thus suppose T=13 when N=:100 U=:21, Equation

(123b) gives

dL ^

or the maximum carrying power would be reached ; all which seems to be in strict

a,ccordance with experience, particularly with new brasses.

42. Application of the Equations to Mr. Tower's Experiments with Brass No. 2 to

determine the Oil Pressure round the JourndL

The approximate equation (74) is available to determine the pressure at any part

of the journal, i.e., for any value of so long as c is small, but these approximations

fail for much smaller values of c than for others, for this reason, together with the fact

that the only case in which the pressure has been measured c is large, the pressures

have only been calculated for c=*5, in which the approximations have been carried to

the extreme extent.

These are obtained directly from equation (107), and the pressures divided by JL-^c

are given in Table II., Section VI.

The results of Mr. Tower's experiments with brass No* 2 are given in Table XII.,

Art. 34.

Had the friction been recorded in the experiments in which Mr. Toweh measured

the pressures with brass No. 2 as with brass No. 1, the values of c might have been

obtained as in the case of brass No. 1. But as this was not done the value of c for

these experiments with brass No. 2 could only be inferred from the agreement of the

relative oil pressures measured in different parts of the journal, those calculated for

the same parts with a particular value of c. This was a matter of trial, and as it was

found that the agreement was very close when

c=*5,

further attempts were not made.

230 PROFESSOB O. REYNOLDS ON THE THEORY OF LUBRICATION

With the section at the middle of the brass the calculated and experimental results

are shown in Table V.

Table V.—Comparison of Relative Pressures, calculated by Equation (107) when

c=*5, with the Pressures Measured by Mr. Tower, see Table XII._, Art. 34,

Brass No. 2.

The values of 9

measured from middleof arc at whichpressures were

measured.

Pressure measuredat the aiiddle of

the journal.

Table XIL, Tower.

P-Po

calculated.

Table II.

Relative values,

experimental.

Relative values,

calculated.-Kic.

o / //

29 20 20

29 20 20

500625370

•7923

1-0150

•6609

•800

1-000

•592

•781

1^000

•651

639615560

This agreement, although not exact, is, considering the nature of the test, very close.

The divergence seems to show that in the experiments c was somewhat more than '5,

but it is doubtful if the agreement would have been exact as, owing to the journal

having been run in one direction only, it seems probable that the radius of the brass

was probably slightly greatest on the on side.

Deducing the value of K^c by dividing the experimental pressure by the calculated

values of ~^~ the values pfiven in the last column are found. An alteration in the value

of c would but slightly have altered the middle value of ^^-—- in the same direction as the

alteration of c; hence taking this value, and making c=*520, as being nearer the real

value,

KiC=-640 .......... (159)

In these experiments

N= 150

From equation (104)

JU "-^ ooo • . . «

L=-2-5504xKcL . ^ ^-==4084

(160)

(161)

therefore

L4L'

= 1-21

To find a

(162)

K, 1230


by equation (150)66-086-0665T„.

N

whence

a/=:000001088e-^^^^^''^ (163)

and taking T^ tbe same as with brass No. 1, and olive oil at N=1805 '^•^*> 5*83° Fahr.,

with brass No. 2, at 70*5^ Fahr.

a^=00086^

instead of with brass No. 1 |> (164)

a,,===00077

The difference in the radii of curvature of the two brasses, the one deduced from

the measured friction, the other deduced from the measured differences of pressure at

different positions round the journal, come out equal within xo^ooo^^-h P^i^t of an inch,

and the values of a differing only by 11 per cent. Had the frictions been given with

brass No. 2, this agreement would have afforded an independent comparison of the

values of a. As it is, the only probability of equality in these two brasses arises from

the probability of their having been bedded in the same way.

In deducing the value a for brass No. 2, it has been assumed that the oil, which was

mineral, had the same law of viscosity as the olive oil. Both these oils were used with

brass No. 1, and the results are nearly the same, the mean resistances, as given by

Mr. Tower, are as 0*623 to 0*654, or the viscosity of the mineral oil being 0*95 that

for olive oil ; had this been taken into account, the value of a^ for brass No. 2 would

have been still nearer that for brass No. 1, being '00084 as against '00077.

As the radii of the two brasses seem to be so near, and as the resistance was

measured for brass No. 1 under circumstances closely resembling those of the

experiment with No. 2, a further test of the exactness of the theory is furnished by

comparing the calculated friction w^th brass No. 2 with that measured with brass

No. 1, with the same oil, the same speed, and nearly the same load.

As in equation (158)

-/' :=:2'3773Kg+l•37(7^— 1)JC^ ,,,..,. (165)

-K.=:'21^N^ a

Whence, taking account of the values of /x for mineral and olive oils, and the values

of a for brass No. 1 and No. 2 for mineral oil and brass No. 2, K2 has 0*871 of the

value in equation (150)

•871X -00346-04^3^.^ , .

232 PBOFESSOB 0. RBYFOLDS OF THE THEORY OP LUBRIOATIOE"

which, when T=5'83'', N~150, L'=337, n=l*25, being substituted in equation

K3=:-^0-1665

In Mr. Tower's Table IV., it appears that with brass No. 1, mineral oil,

N=150 L'=:310 f=i'4: L'=U5 f='51

whence interpolating for

L'=:337

/=:4-58. ..,....,.. (168)

This agreement is verj close, for taking account of the difference of radius, the

calculated friction for brass No. 2 should have been about *95 of the measured friction

with brass No, 2.

In order to show the agreement between the calculated pressures and tbos^ of Mr.

Tower, the values of "^^^ for c=-5 have been plotted, and are shown in Plate 8,

figs, 2 and 3, the crosses indicating the experiments with brass No, 2, as in Table VII.

(Tower),

43. Conclusions,

The experiments to which the theory has been definitely applied rnay be taken to

include all Mr. Tower's experiments with the 4-inch journal and oil bath, in which

the number of revolutions per minute was between 100 and 450, and the nominal

loads in lbs. per sq. inch between 100 and 415, The other experiments with the oil

bath were with loads from 415, till the journal seized at 520, 573, or 625 ; and a set

of experiments with brass No. 2 at 20 revolutions per minute. All these experiments

were under extreme conditions, for which, by the theory, c was so great as to render

lubrication incomplete, and preclude the application of the theory without further

integrations.

The theory has, therefore, been tested by experiments throughout the entire range

of circumstances to which the particular integrations undertaken are applicable. Andthe resr^lts, which in many cases check one another, are consistent throughout.

The agreement of the experimental results with the particular equations obtained

on the assumption that the brass, as well as the journal, are truly circular, must be

attributed to the same causes as the great regularity presented by the experimental

results themselves.

Fundamental amongst these causes is, as Mr. ToWEH has pointed out, the perfect

Bupply of lubricant obtained with the oil bath. But scarcely less important must

have been the truth with which the brasses were first fitted to the journal, the

AKD ITS APPLICATION TO MR. B. TOWER'S EXPERIMENTS. 233

smallness of the subsequent wear, and the variety of the conditions as to magnitude

of load, speed, and direction of motion.

That a brass in continuous use should preserve a circular section with a constant

radius requires either that there should be no wear at all, or that the wear at any

point P should be proportional to sin (90°--POH).

Experience shows that there is wear in ordinary practice, and even in Mr. TowER^s

experiments there seems to have been some wear. In these experiments, however,

there is every reason to suppose that the wear would have been approximately pro-

portional to c sin (<^Q— ^)= c sin (90°—POH), because this represents the approach of

the brass to the journal within the mean distance a, for all points, except those at

which it is negative ; at these there would be no wear. So long then as the journal

ran in one direction only, the wear would tend to preserve the radius and true circular

form of that portion of the arc from C to F (fig. 17, p. 193), altering the radius at F,

and enlarging it from F to D. On reversal, however, C and F change sides, and hence

alternate motion in both directions would preserve the radius constant all over the

brass. The experience emphasized by Mr. Toweh that the journal after running for

some time in one direction would not run at first in the other, strongly bears out this

conclusion. Hence it follows that had the journal been continuously run in one

direction the condition of lubrication, as shown by the distribution of oil pressure

round the journal, would have been modified, the pressure falling between O and Bon the on side of the journal, a conclusion which is borne out by the fact that in the

experiments with brass No. 2, which was run for some time continuously in one

direction, the pressure measured on the on side is somewhat below that calculated on

the assumption of circular form, although the agreement is close for the other four

points (see fig. 2, Plate 8).

When the surfaces are completely separated by oil it is difficult to see what can

cause wear. But there is generally metallic contact at starting, and hence abrasion,

which will introduce metallic particles into the oil (blacken it) ; these particles will be

more or less carried round and round, causing wear and increasing the number of

particles and the viscosity of the oil. Thus the rate of wear would depend on the

1impurities in the oil, the values of c, - and the velocity of the journal, and hence

CO

would render the greatest velocity at which the maximum load could be carried with

a large value of c small. A conclusion which seems to be confirmed by Mr. Tower's

experiments at twenty revolutions per minute.

In cases such as engine bearings the wear causes the radius of curvature of the

brass continually to increase, and hence a and c must continually increase with wear.

But in order to apply the theory to such cases the change in the direction of the

load (or U^ and V^) have to be taken into account.

That the circumstances of Mr. Tower's experiments are not those of ordinary

practice, and hence that the particular equations deduced in order to apply the theory

MDCCCLXXXVI. 2 H

234 PEOFESSOR 0. REYN^OLDS ON" THE THEORY OF LUBRICATION, ETC.

to these experiments do not apply to ofdinarj cases, does not show that the general

theory, as given in equations (15), (18), and (19), could not be applied to ordinary

cases were the conditions sufficiently known.

These experiments of Mr. Tower have afforded the means of verifying the theory

for a particular case, and hence has established its truth as applicable to all cases for

which the integrations can be effected.

The circumstances expressed by

L a^' U' R' ^' ^^' ^1' ^^' ^' ^' ^' ^' ^'

which are shown by the theory to be the principal circumstances on which lubrication

depends, although not the same in other cases, will still be the principal circumstances

and indicate the conditions to be fulfilled in order to secure good lubrication.

The verification of the equations for viscous fluids under such extreme circum-

stances affords a severe test of the truth and completeness of the assumptions on

which these eauations were founded. While the result of the whole research is to

point to a conclusion (important in Natural Philosophy) that not only in cases of

intentional lubrication, but wherever hard surfaces under pressure slide over each

other without abrasion, they are separated by a film of some foreign matter whether

perceivable or not. And that the question as to whether this action can be con-

tinuous or not turns on whether the action tends to preserve the matter between the

surfaces at the points of pressure as in the apparently unique case of the revolving

journal, or tends to sweep it to one side as is the result of all backwards and forward

rubbing with continuous pressure.

The fact that a little grease will enable almost any surfaces to slide for a time has

tended doubtless to obscure the action of the revolving journal to maintain the oil

between the surfaces at the point of pressure. And yet, although only now under-

stood, it is this action that has alone rendered our machines, and even our carriages

possible. The only other self-acting system of lubrication is that of reciprocating

joints with alternate pressure on and separation (drawing the oil back or a fresh

supplj^) of the surfaces. This plays an important part in certain machines, as in

the steam engine, and is as fundamental to animal mechanics as the lubricating action

of the journal is to mechanical contrivances.

0/Reynolds. Phil. Trans. \SS^. Plate 8.

Fi g . 3 Fig. 2

::,

i /^

^'-!i : I-'

''.\\ i :

f.O

0.9i

0.8\

01

!I

. / i

'

i

I '

\

_.^__J_„..i L/ I .:

'J?^TO ZO SO 40 50 60 70 [SO

Fig ,

4*.

Fig.l

West,lTewi^n.sai X-Co.sc.

by osborne reynolds, ll,d,, fm.s. contents. · 2011. 8. 2. ·...

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